{"history":[{"id":"14532de2-b6cb-4c44-9bb4-3e67c5ab0fe1","wordChanged":"**Fourier transform**","newWord":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), the SigmaGyatt (**FT**) is an [integral transform](https://bliptext.com/articles/integral-transform) that takes a [function](https://bliptext.com/articles/function-mathematics) as input and outputs another [function](https://bliptext.com/articles/function-mathematics) that describes the extent to which various frequencies are present in the original [function](https://bliptext.com/articles/function-mathematics).\nThe output of the transform is a [complex](https://bliptext.com/articles/complex-number)-valued function of frequency.\nThe term Fourier transform refers to both this complex-valued function and the [mathematical operation](https://bliptext.com/articles/operation-mathematics).\nWhen a distinction needs to be made, the output of the operation is sometimes called the [frequency domain](https://bliptext.com/articles/frequency-domain) representation of the original function.\nThe Fourier transform is analogous to decomposing the [sound](https://bliptext.com/articles/sound) of a musical [chord](https://bliptext.com/articles/chord-music) into the [intensities](https://bliptext.com/articles/sound-intensity) of its constituent [pitches](https://bliptext.com/articles/pitch-music).\nFunctions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa, a phenomenon known as the [uncertainty principle](https://bliptext.com/articles/).\nThe [critical](https://bliptext.com/articles/critical-point-mathematics) case for this principle is the [Gaussian function](https://bliptext.com/articles/gaussian-function), of substantial importance in [probability theory](https://bliptext.com/articles/probability-theory) and [statistics](https://bliptext.com/articles/statistics) as well as in the study of physical phenomena exhibiting [normal distribution](https://bliptext.com/articles/normal-distribution) (e.g., [diffusion](https://bliptext.com/articles/diffusion)).\nThe Fourier transform of a Gaussian function is another Gaussian function.\n[Joseph Fourier](https://bliptext.com/articles/joseph-fourier) introduced [sine and cosine transforms](https://bliptext.com/articles/sine-and-cosine-transforms) (which [correspond to the imaginary and real components](https://bliptext.com/articles/sine-and-cosine-transforms) of the modern Fourier transform) in his study of [heat transfer](https://bliptext.com/articles/heat-transfer), where Gaussian functions appear as solutions of the [heat equation](https://bliptext.com/articles/heat-equation).\nThe Fourier transform can be formally defined as an [improper](https://bliptext.com/articles/improper-integral) [Riemann integral](https://bliptext.com/articles/riemann-integral), making it an integral transform, although this definition is not suitable for many applications requiring a more sophisticated integration theory.\nFor example, many relatively simple applications use the [Dirac delta function](https://bliptext.com/articles/dirac-delta-function), which can be treated formally as if it were a function, but the justification requires a mathematically more sophisticated viewpoint.\nThe Fourier transform can also be generalized to functions of several variables on [Euclidean space](https://bliptext.com/articles/euclidean-space), sending a function of 3-dimensional 'position space' to a function of 3-dimensional momentum (or a function of space and time to a function of [4-momentum](https://bliptext.com/articles/4-momentum)).\nThis idea makes the spatial Fourier transform very natural in the study of waves, as well as in [quantum mechanics](https://bliptext.com/articles/quantum-mechanics), where it is important to be able to represent wave solutions as functions of either position or momentum and sometimes both.\nIn general, functions to which Fourier methods are applicable are complex-valued, and possibly [vector-valued](https://bliptext.com/articles/vector-valued-function).\nStill further generalization is possible to functions on [groups](https://bliptext.com/articles/group-mathematics), which, besides the original Fourier transform on [[Real number#Arithmetic|\nR\n]] or\nRn\n, notably includes the [discrete-time Fourier transform](https://bliptext.com/articles/discrete-time-fourier-transform) (DTFT, group =\nZ\n), the [discrete Fourier transform](https://bliptext.com/articles/discrete-fourier-transform) (DFT, group = [[cyclic group|\nZ mod N\n]]) and the [Fourier series](https://bliptext.com/articles/fourier-series) or circular Fourier transform (group =\nS1\n, the unit circle ≈ closed finite interval with endpoints identified). The latter is routinely employed to handle [periodic functions](https://bliptext.com/articles/periodic-function). The [fast Fourier transform](https://bliptext.com/articles/fast-fourier-transform) (FFT) is an algorithm for computing the DFT.\n# [An example application of the Fourier transform is determining the constituent pitches in a musical waveform. This image is the result of applying a constant-Q transform (a Fourier-related transform) to the waveform of a C major piano chord. The first three peaks on the left correspond to the frequencies of the fundamental frequency of the chord (C, E, G). The remaining smaller peaks are higher-frequency overtones of the fundamental pitches. A pitch detection algorithm could use the relative intensity of these peaks to infer which notes the pianist pressed. | upload.wikimedia.org/wikipedia/commons/3/31/CQT-piano-chord.png]\n\n# Definition\nThe Fourier transform is an analysis process, decomposing a complex-valued function into its constituent frequencies and their amplitudes.\nThe inverse process is synthesis, which recreates from its transform.\nWe can start with an analogy, the **[Fourier series](https://bliptext.com/articles/fourier-series)**, which analyzes on a bounded interval for some positive real number P. The constituent frequencies are a discrete set of harmonics at frequencies whose amplitude and phase are given by the **analysis formula****:******The actual **[Fourier series](https://bliptext.com/articles/fourier-series)** is the **synthesis formula****:******On an unbounded interval, P\\to\\infty, the constituent frequencies are a continuum****:**** and c_n is replaced by a function****:****\nEvaluating for all values of \\xi produces the frequency-domain function.\nThe integral can diverge at some frequencies.\n(see ) But it converges for ** all ** frequencies when f(x) decays with** all **derivatives as **:**.\n(See [Schwartz function](https://bliptext.com/articles/schwartz-function)).\nBy the [Riemann–Lebesgue lemma](https://bliptext.com/articles/riemann-lebesgue-lemma), the transformed function \\widehat f also decays with all derivatives.\nThe complex number, in polar coordinates, conveys both [amplitude](https://bliptext.com/articles/amplitude) and [phase](https://bliptext.com/articles/phase-offset) of frequency \\xi.\nThe intuitive interpretation of is that the effect of multiplying f(x) by is to subtract \\xi from every frequency component of function f(x).\nOnly the component that was at frequency \\xi can produce a non-zero value of the infinite integral, because (at least formally) all the other shifted components are oscillatory and integrate to zero.\n(see )\nThe corresponding synthesis formula is:\nis a representation of f(x) as a weighted summation of complex exponential functions.\nThis is also known as the [Fourier inversion theorem](https://bliptext.com/articles/fourier-inversion-theorem), and was first introduced in [Fourier's](https://bliptext.com/articles/joseph-fourier) Analytical Theory of Heat.\nThe functions f and \\widehat{f} are referred to as a **Fourier transform pair**.\nA common notation for designating transform pairs is**:**\nfor example\n\n## Definition for Lebesgue integrable functions\nUntil now, we have been dealing with Schwartz functions, which decay rapidly at infinity, with all derivatives.\nThis excludes many functions of practical importance from the definition, such as the [rect function](https://bliptext.com/articles/rect-function).\nA [measurable function](https://bliptext.com/articles/measurable-function) is called (Lebesgue) integrable if the [Lebesgue integral](https://bliptext.com/articles/lebesgue-integral) of its absolute value is finite:\nTwo measurable functions are equivalent if they are equal except on a set of measure zero.\nThe set of all equivalence classes of integrable functions is denoted.\nThen:\nThe integral is well-defined for all because of the assumption.\n(It can be shown that the function is bounded and [uniformly continuous](https://bliptext.com/articles/uniformly-continuous) in the frequency domain, and moreover, by the [Riemann–Lebesgue lemma](https://bliptext.com/articles/riemann-lebesgue-lemma), it is zero at infinity.)\nHowever, the class of Lebesgue integrable functions is not ideal from the point of view of the Fourier transform because there is no easy characterization of the image, and thus no easy characterization of the inverse transform.\n\n## Unitarity and definition for square integrable functions\nWhile defines the Fourier transform for (complex-valued) functions in, it is easy to see that it is not well-defined for other integrability classes, most importantly.\nFor functions in, and with the conventions of , the Fourier transform is a [unitary operator](https://bliptext.com/articles/unitary-operator) with respect to the Hilbert inner product on , restricted to the dense subspace of integrable functions.\nTherefore, it admits a unique continuous extension to a unitary operator on, also called the Fourier transform.\nThis extension is important in part because the Fourier transform preserves the space so that, unlike the case of L^1, the Fourier transform and inverse transform are on the same footing, being transformations of the same space of functions to itself.\nImportantly, for functions in L^2, the Fourier transform is no longer given by (interpreted as a Lebesgue integral).\nFor example, the function is in L^2 but not L^1, so the integral diverges.\nIn such cases, the Fourier transform can be obtained explicitly by regularizing the integral, and then passing to a limit.\nIn practice, the integral is often regarded as an [improper integral](https://bliptext.com/articles/improper-integral) instead of a proper Lebesgue integral, but sometimes for convergence one needs to use [weak limit](https://bliptext.com/articles/weak-limit) or [principal value](https://bliptext.com/articles/cauchy-principal-value) instead of the (pointwise) limits implicit in an [improper integral](https://bliptext.com/articles/improper-integral).\nand each gives three rigorous ways of extending the Fourier transform to square integrable functions using this procedure.\nThe conventions chosen in this article are those of [harmonic analysis](https://bliptext.com/articles/harmonic-analysis), and are characterized as the unique conventions such that the Fourier transform is both [unitary](https://bliptext.com/articles/unitary-operator) on\nL2\nand an algebra homomorphism from\nL1\nto\nL∞\n, without renormalizing the Lebesgue measure.\n\n## Angular frequency (ω)\nWhen the independent variable (x) represents time (often denoted by t), the transform variable (\\xi) represents [frequency](https://bliptext.com/articles/frequency) (often denoted by f).\nFor example, if time is measured in [seconds](https://bliptext.com/articles/second), then frequency is in [hertz](https://bliptext.com/articles/hertz).\nThe Fourier transform can also be written in terms of [angular frequency](https://bliptext.com/articles/angular-frequency), whose units are [radians](https://bliptext.com/articles/radian) per second.\nThe substitution into produces this convention, where function \\widehat f is relabeled\nUnlike the definition, the Fourier transform is no longer a [unitary transformation](https://bliptext.com/articles/unitary-transformation), and there is less symmetry between the formulas for the transform and its inverse.\nThose properties are restored by splitting the 2 \\pi factor evenly between the transform and its inverse, which leads to another convention:\nVariations of all three conventions can be created by conjugating the complex-exponential [kernel](https://bliptext.com/articles/integral-kernel) of both the forward and the reverse transform.\nThe signs must be opposites.\n\n## Extension of the definition\nFor 1 0\nsuch that for any [integer](https://bliptext.com/articles/integer)\nn ≥ 0\n,\nfor some constant C. (In this case, f is supported on\n[−a, a]\n.) This can be expressed by saying that\nf̂\nis an [entire function](https://bliptext.com/articles/entire-function) which is [rapidly decreasing](https://bliptext.com/articles/rapidly-decreasing) in σ (for fixed τ) and of exponential growth in τ (uniformly in σ).\n(If f is not smooth, but only\nL2\n, the statement still holds provided\n. ) The space of such functions of a [complex variable](https://bliptext.com/articles/complex-analysis) is called the Paley—Wiener space. This theorem has been generalised to semisimple [Lie groups](https://bliptext.com/articles/lie-group).\nIf f is supported on the half-line\nt ≥ 0\n, then f is said to be \"causal\" because the [impulse response function](https://bliptext.com/articles/impulse-response-function) of a physically realisable [filter](https://bliptext.com/articles/filter-mathematics) must have this property, as no effect can precede its cause.\n[Paley](https://bliptext.com/articles/raymond-paley) and Wiener showed that then\nf̂\nextends to a [holomorphic function](https://bliptext.com/articles/holomorphic-function) on the complex lower half-plane\nτ < 0\nwhich tends to zero as τ goes to infinity.\nThe converse is false and it is not known how to characterise the Fourier transform of a causal function.\n\n## Laplace transform\nThe Fourier transform\nf̂(ξ)\nis related to the [Laplace transform](https://bliptext.com/articles/laplace-transform)\nF(s)\n, which is also used for the solution of [differential equations](https://bliptext.com/articles/differential-equation) and the analysis of [filters](https://bliptext.com/articles/filter-signal-processing).\nIt may happen that a function f for which the Fourier integral does not converge on the real axis at all, nevertheless has a complex Fourier transform defined in some region of the [complex plane](https://bliptext.com/articles/complex-plane).\nFor example, if\nf(t)\nis of exponential growth, i.e.,\nfor some constants\nC, a ≥ 0\n, then\nconvergent for all\n2πτ < −a\n, is the [two-sided Laplace transform](https://bliptext.com/articles/two-sided-laplace-transform) of f.\nThe more usual version (\"one-sided\") of the Laplace transform is\nIf f is also causal, and analytical, then: Thus, extending the Fourier transform to the complex domain means it includes the Laplace transform as a special case in the case of causal functions—but with the change of variable\n.\nFrom another, perhaps more classical viewpoint, the Laplace transform by its form involves an additional exponential regulating term which lets it converge outside of the imaginary line where the Fourier transform is defined.\nAs such it can converge for at most exponentially divergent series and integrals, whereas the original Fourier decomposition cannot, enabling analysis of systems with divergent or critical elements.\nTwo particular examples from linear signal processing are the construction of allpass filter networks from critical comb and mitigating filters via exact pole-zero cancellation on the unit circle.\nSuch designs are common in audio processing, where highly nonlinear phase response is sought for, as in reverb.\nFurthermore, when extended pulselike impulse responses are sought for signal processing work, the easiest way to produce them is to have one circuit which produces a divergent time response, and then to cancel its divergence through a delayed opposite and compensatory response.\nThere, only the delay circuit in-between admits a classical Fourier description, which is critical.\nBoth the circuits to the side are unstable, and do not admit a convergent Fourier decomposition.\nHowever, they do admit a Laplace domain description, with identical half-planes of convergence in the complex plane (or in the discrete case, the Z-plane), wherein their effects cancel.\nIn modern mathematics the Laplace transform is conventionally subsumed under the aegis Fourier methods.\nBoth of them are subsumed by the far more general, and more abstract, idea of [harmonic analysis](https://bliptext.com/articles/harmonic-analysis).\n\n## Inversion\nStill with, if \\widehat f is complex analytic for\na ≤ τ ≤ b\n, then\nby [Cauchy's integral theorem](https://bliptext.com/articles/cauchy-s-integral-theorem).\nTherefore, the Fourier inversion formula can use integration along different lines, parallel to the real axis.\nTheorem: If\nf(t) = 0\nfor\nt < 0\n, and\nfor some constants\nC, a > 0\n, then\nfor any\nτ < −a⁄2π\n.\nThis theorem implies the [Mellin inversion formula](https://bliptext.com/articles/inverse-laplace-transform) for the Laplace transformation,\nfor any\nb > a\n, where\nF(s)\nis the Laplace transform of\nf(t)\n.\nThe hypotheses can be weakened, as in the results of Carleson and Hunt, to\nf(t) e−at\nbeing\nL1\n, provided that f be of bounded variation in a closed neighborhood of t (cf. [Dini test](https://bliptext.com/articles/dini-test)), the value of f at t be taken to be the [arithmetic mean](https://bliptext.com/articles/arithmetic-mean) of the left and right limits, and that the integrals be taken in the sense of Cauchy principal values.\nL2\nversions of these inversion formulas are also available.\n\n# Fourier transform on Euclidean space\nThe Fourier transform can be defined in any arbitrary number of dimensions n. As with the one-dimensional case, there are many conventions.\nFor an integrable function\nf(x)\n, this article takes the definition:\nwhere\nx\nand\nξ\nare n-dimensional [vectors](https://bliptext.com/articles/vector-mathematics), and\nx · ξ\nis the [dot product](https://bliptext.com/articles/dot-product) of the vectors.\nAlternatively,\nξ\ncan be viewed as belonging to the [dual vector space](https://bliptext.com/articles/dual-space) \\R^{n\\star}, in which case the dot product becomes the [contraction](https://bliptext.com/articles/tensor-contraction) of\nx\nand\nξ\n, usually written as\n⟨x, ξ⟩\n.\nAll of the basic properties listed above hold for the n-dimensional Fourier transform, as do Plancherel's and Parseval's theorem.\nWhen the function is integrable, the Fourier transform is still uniformly continuous and the [Riemann–Lebesgue lemma](https://bliptext.com/articles/riemann-lebesgue-lemma) holds.\n\n## Uncertainty principle\nGenerally speaking, the more concentrated\nf(x)\nis, the more spread out its Fourier transform\nf̂(ξ)\nmust be.\nIn particular, the scaling property of the Fourier transform may be seen as saying: if we squeeze a function in x, its Fourier transform stretches out in ξ.\nIt is not possible to arbitrarily concentrate both a function and its Fourier transform.\nThe trade-off between the compaction of a function and its Fourier transform can be formalized in the form of an [uncertainty principle](https://bliptext.com/articles/uncertainty-principle) by viewing a function and its Fourier transform as [conjugate variables](https://bliptext.com/articles/conjugate-variables) with respect to the [[symplectic form](https://bliptext.com/articles/symplectic-vector-space)](https://bliptext.com/articles/symplectic-form) on the [time–frequency domain](https://bliptext.com/articles/time-frequency-representation): from the point of view of the [linear canonical transformation](https://bliptext.com/articles/linear-canonical-transformation), the Fourier transform is rotation by 90° in the [time–frequency domain](https://bliptext.com/articles/time-frequency-representation), and preserves the [[symplectic form](https://bliptext.com/articles/symplectic-vector-space)](https://bliptext.com/articles/symplectic-form).\nSuppose\nf(x)\nis an integrable and [square-integrable](https://bliptext.com/articles/square-integrable) function.\nWithout loss of generality, assume that\nf(x)\nis normalized:\nIt follows from the [Plancherel theorem](https://bliptext.com/articles/plancherel-theorem) that\nf̂(ξ)\nis also normalized.\nThe spread around\nmay be measured by the dispersion about zero defined by\nIn probability terms, this is the [second moment](https://bliptext.com/articles/moment-mathematics) of\nabout zero.\nThe uncertainty principle states that, if\nf(x)\nis absolutely continuous and the functions\nx·f(x)\nand\nf(x)\nare square integrable, then\nThe equality is attained only in the case\nwhere\nσ > 0\nis arbitrary and\nC1 = 4√2⁄√σ\nso that f is\nL2\n-normalized.\nIn other words, where f is a (normalized) [Gaussian function](https://bliptext.com/articles/gaussian-function) with variance\nσ2/2π\n, centered at zero, and its Fourier transform is a Gaussian function with variance\nσ−2/2π\n.\nIn fact, this inequality implies that:\nfor any\nx0\n,\nξ0 ∈ R\n.\nIn [quantum mechanics](https://bliptext.com/articles/quantum-mechanics), the [momentum](https://bliptext.com/articles/momentum) and position [wave functions](https://bliptext.com/articles/wave-function) are Fourier transform pairs, up to a factor of the [Planck constant](https://bliptext.com/articles/planck-constant).\nWith this constant properly taken into account, the inequality above becomes the statement of the [Heisenberg uncertainty principle](https://bliptext.com/articles/heisenberg-uncertainty-principle).\nA stronger uncertainty principle is the [Hirschman uncertainty principle](https://bliptext.com/articles/hirschman-uncertainty), which is expressed as:\nwhere\nH(p)\nis the [differential entropy](https://bliptext.com/articles/differential-entropy) of the [probability density function](https://bliptext.com/articles/probability-density-function)\np(x)\nwhere the logarithms may be in any base that is consistent.\nThe equality is attained for a Gaussian, as in the previous case.\n\n## Sine and cosine transforms\nFourier's original formulation of the transform did not use complex numbers, but rather sines and cosines.\nStatisticians and others still use this form.\nAn absolutely integrable function f for which Fourier inversion holds can be expanded in terms of genuine frequencies (avoiding negative frequencies, which are sometimes considered hard to interpret physically ) λ by\nThis is called an expansion as a trigonometric integral, or a Fourier integral expansion.\nThe coefficient functions a and b can be found by using variants of the Fourier cosine transform and the Fourier sine transform (the normalisations are, again, not standardised):\nand\nOlder literature refers to the two transform functions, the Fourier cosine transform, a, and the Fourier sine transform, b.\nThe function f can be recovered from the sine and cosine transform using\ntogether with trigonometric identities.\nThis is referred to as Fourier's integral formula.\n\n## Spherical harmonics\nLet the set of [homogeneous](https://bliptext.com/articles/homogeneous-polynomial) [harmonic](https://bliptext.com/articles/harmonic-function) [polynomials](https://bliptext.com/articles/polynomial) of degree k on\nRn\nbe denoted by\nAk\n. The set\nAk\nconsists of the [solid spherical harmonics](https://bliptext.com/articles/solid-spherical-harmonics) of degree k. The [solid spherical harmonics](https://bliptext.com/articles/solid-spherical-harmonics) play a similar role in higher dimensions to the Hermite polynomials in dimension one.\nSpecifically, if\nf(x) = e−π P(x)\nfor some\nP(x)\nin\nAk\n, then\nf̂(ξ) = i f(ξ)\n. Let the set\nHk\nbe the closure in\nL2(Rn)\nof linear combinations of functions of the form\nf(\nwhere\nP(x)\nis in\nAk\n. The space\nL2(Rn)\nis then a direct sum of the spaces\nHk\nand the Fourier transform maps each space\nHk\nto itself and is possible to characterize the action of the Fourier transform on each space\nHk\n.\nLet\n)P(x)\n(with\nP(x)\nin\nAk\n), then\nwhere\nHere\nJ(n + 2k − 2)/2\ndenotes the [Bessel function](https://bliptext.com/articles/bessel-function) of the first kind with order\nn + 2k − 2⁄2\n. When\nthis gives a useful formula for the Fourier transform of a radial function.\nThis is essentially the [Hankel transform](https://bliptext.com/articles/hankel-transform).\nMoreover, there is a simple recursion relating the cases\nn + 2\nand n allowing to compute, e.g., the three-dimensional Fourier transform of a radial function from the one-dimensional one.\n\n## Restriction problems\nIn higher dimensions it becomes interesting to study restriction problems for the Fourier transform.\nThe Fourier transform of an integrable function is continuous and the restriction of this function to any set is defined.\nBut for a square-integrable function the Fourier transform could be a general class of square integrable functions.\nAs such, the restriction of the Fourier transform of an\nL2(Rn)\nfunction cannot be defined on sets of measure 0.\nIt is still an active area of study to understand restriction problems in\nL\nfor\n1 < p < 2\n. It is possible in some cases to define the restriction of a Fourier transform to a set S, provided S has non-zero curvature.\nThe case when S is the unit sphere in\nRn\nis of particular interest.\nIn this case the Tomas–[Stein](https://bliptext.com/articles/elias-stein) restriction theorem states that the restriction of the Fourier transform to the unit sphere in\nRn\nis a bounded operator on\nL\nprovided\n1 ≤ p ≤ 2n + 2⁄n + 3\n.\nOne notable difference between the Fourier transform in 1 dimension versus higher dimensions concerns the partial sum operator.\nConsider an increasing collection of measurable sets\nER\nindexed by\nR ∈ (0,∞)\n2R\n. For a given integrable function f, consider the function fR defined by:\nSuppose in addition that\nf ∈ L(Rn)\n. For\nand\n1 < p < ∞\n, if one takes\n, then fR converges to f in\nL\nas R tends to infinity, by the boundedness of the [Hilbert transform](https://bliptext.com/articles/hilbert-transform).\nNaively one may hope the same holds true for\nn > 1\n. In the case that ER is taken to be a cube with side length R, then convergence still holds.\nAnother natural candidate is the Euclidean ball {{math|E{{sub|R}} = {ξ : {{abs|ξ}} < R{{)}}}}. In order for this partial sum operator to converge, it is necessary that the multiplier for the unit ball be bounded in\nL(Rn)\n. For\nn ≥ 2\nit is a celebrated theorem of [Charles Fefferman](https://bliptext.com/articles/charles-fefferman) that the multiplier for the unit ball is never bounded unless {{math|p {{=}} 2}}.\nIn fact, when\np ≠ 2\n, this shows that not only may fR fail to converge to f in\nL\n, but for some functions\nf ∈ L(Rn)\n, fR is not even an element of\nL\n.\n\n# Fourier transform on function spaces\n\n## On Lp spaces\n\n### On L1\nThe definition of the Fourier transform by the integral formula\nis valid for Lebesgue integrable functions f; that is,\nf ∈ L1(Rn)\n.\nThe Fourier transform\nis a [bounded operator](https://bliptext.com/articles/bounded-operator).\nThis follows from the observation that\nwhich shows that its [operator norm](https://bliptext.com/articles/operator-norm) is bounded by 1.\nIndeed, it equals 1, which can be seen, for example, from the [transform of the rect function](https://bliptext.com/articles/).\nThe image of\nL1\nis a subset of the space\nC0(Rn)\nof continuous functions that tend to zero at infinity (the [Riemann–Lebesgue lemma](https://bliptext.com/articles/riemann-lebesgue-lemma)), although it is not the entire space.\nIndeed, there is no simple characterization of the image.\n\n### On L2\nSince compactly supported smooth functions are integrable and dense in\nL2(Rn)\n, the [Plancherel theorem](https://bliptext.com/articles/plancherel-theorem) allows one to extend the definition of the Fourier transform to general functions in\nL2(Rn)\nby continuity arguments.\nThe Fourier transform in\nL2(Rn)\nis no longer given by an ordinary Lebesgue integral, although it can be computed by an [improper integral](https://bliptext.com/articles/improper-integral), here meaning that for an\nL2\nfunction f,\nwhere the limit is taken in the\nL2\nsense.\n)\nMany of the properties of the Fourier transform in\nL1\ncarry over to\nL2\n, by a suitable limiting argument.\nFurthermore,\nis a [unitary operator](https://bliptext.com/articles/unitary-operator).\nFor an operator to be unitary it is sufficient to show that it is bijective and preserves the inner product, so in this case these follow from the Fourier inversion theorem combined with the fact that for any\nf, g ∈ L2(Rn)\nwe have\nIn particular, the image of\nL2(Rn)\nis itself under the Fourier transform.\n\n### On other Lp\nThe definition of the Fourier transform can be extended to functions in\nL(Rn)\nfor\n1 ≤ p ≤ 2\nby decomposing such functions into a fat tail part in\nL2\nplus a fat body part in\nL1\n. In each of these spaces, the Fourier transform of a function in\nL(Rn)\nis in\nL(Rn)\n, where\nq = p⁄p − 1\nis the [Hölder conjugate](https://bliptext.com/articles/h-lder-conjugate) of p (by the [Hausdorff–Young inequality](https://bliptext.com/articles/hausdorff-young-inequality)).\nHowever, except for\np = 2\n, the image is not easily characterized.\nFurther extensions become more technical.\nThe Fourier transform of functions in\nL\nfor the range\n2 < p < ∞\nrequires the study of distributions.\nIn fact, it can be shown that there are functions in\nL\nwith\np > 2\nso that the Fourier transform is not defined as a function.\n\n## Tempered distributions\nOne might consider enlarging the domain of the Fourier transform from\nL1 + L2\nby considering [generalized functions](https://bliptext.com/articles/generalized-function), or distributions.\nA distribution on\nRn\nis a continuous linear functional on the space\nCc(Rn)\nof compactly supported smooth functions, equipped with a suitable topology.\nThe strategy is then to consider the action of the Fourier transform on\nCc(Rn)\nand pass to distributions by duality.\nThe obstruction to doing this is that the Fourier transform does not map\nCc(Rn)\nto\nCc(Rn)\n. In fact the Fourier transform of an element in\nCc(Rn)\ncan not vanish on an open set; see the above discussion on the uncertainty principle.\nThe right space here is the slightly larger space of [Schwartz functions](https://bliptext.com/articles/schwartz-space).\nThe Fourier transform is an automorphism on the Schwartz space, as a topological vector space, and thus induces an automorphism on its dual, the space of tempered distributions.\nThe tempered distributions include all the integrable functions mentioned above, as well as well-behaved functions of polynomial growth and distributions of compact support.\nFor the definition of the Fourier transform of a tempered distribution, let f and g be integrable functions, and let\nf̂\nand\nĝ\nbe their Fourier transforms respectively.\nThen the Fourier transform obeys the following multiplication formula,\nEvery integrable function f defines (induces) a distribution Tf by the relation\nfor all Schwartz functions φ.\nSo it makes sense to define Fourier transform\nT̂f\nof\nTf\nby\nfor all Schwartz functions φ.\nExtending this to all tempered distributions T gives the general definition of the Fourier transform.\nDistributions can be differentiated and the above-mentioned compatibility of the Fourier transform with differentiation and convolution remains true for tempered distributions.\n\n# Generalizations\n\n## Fourier–Stieltjes transform\nThe Fourier transform of a [finite](https://bliptext.com/articles/finite-measure) [Borel measure](https://bliptext.com/articles/borel-measure) μ on\nRn\nis given by:\nThis transform continues to enjoy many of the properties of the Fourier transform of integrable functions.\nOne notable difference is that the [Riemann–Lebesgue lemma](https://bliptext.com/articles/riemann-lebesgue-lemma) fails for measures.\nIn the case that\n, then the formula above reduces to the usual definition for the Fourier transform of f. In the case that μ is the probability distribution associated to a random variable X, the Fourier–Stieltjes transform is closely related to the [characteristic function](https://bliptext.com/articles/characteristic-function-probability-theory), but the typical conventions in probability theory take\neiξx\ninstead of\ne−i2πξx\n. In the case when the distribution has a [probability density function](https://bliptext.com/articles/probability-density-function) this definition reduces to the Fourier transform applied to the [probability density function](https://bliptext.com/articles/probability-density-function), again with a different choice of constants.\nThe Fourier transform may be used to give a characterization of measures.\n[Bochner's theorem](https://bliptext.com/articles/bochner-s-theorem) characterizes which functions may arise as the Fourier–Stieltjes transform of a positive measure on the circle.\nFurthermore, the [Dirac delta function](https://bliptext.com/articles/dirac-delta-function), although not a function, is a finite Borel measure.\nIts Fourier transform is a constant function (whose specific value depends upon the form of the Fourier transform used).\n\n## Locally compact abelian groups\nThe Fourier transform may be generalized to any locally compact abelian group.\nA [locally compact](https://bliptext.com/articles/locally-compact) [abelian group](https://bliptext.com/articles/abelian-group) is an [abelian group](https://bliptext.com/articles/abelian-group) that is at the same time a [locally compact](https://bliptext.com/articles/locally-compact) [Hausdorff topological space](https://bliptext.com/articles/hausdorff-space) so that the group operation is continuous.\nIf G is a locally compact abelian group, it has a translation invariant measure μ, called [Haar measure](https://bliptext.com/articles/haar-measure).\nFor a locally compact abelian group G, the set of irreducible, i.e. one-dimensional, unitary representations are called its [characters](https://bliptext.com/articles/character-group).\nWith its natural group structure and the topology of uniform convergence on compact sets (that is, the topology induced by the [compact-open topology](https://bliptext.com/articles/compact-open-topology) on the space of all continuous functions from G to the [circle group](https://bliptext.com/articles/circle-group)), the set of characters Ĝ is itself a locally compact abelian group, called the Pontryagin dual of G. For a function f in\nL1(G)\n, its Fourier transform is defined by\nThe Riemann–Lebesgue lemma holds in this case;\nf̂(ξ)\nis a function vanishing at infinity on Ĝ.\nThe Fourier transform on is an example; here T is a locally compact abelian group, and the Haar measure μ on T can be thought of as the Lebesgue measure on [0,1). Consider the representation of T on the complex plane C that is a 1-dimensional complex vector space. There are a group of representations (which are irreducible since C is 1-dim) where for x\\in T.\nThe character of such representation, that is the trace of e_{k}(x) for each x\\in T and k\\in Z, is itself.\nIn the case of representation of finite group, the character table of the group G are rows of vectors such that each row is the character of one irreducible representation of G, and these vectors form an orthonormal basis of the space of class functions that map from G to C by Schur's lemma.\nNow the group T is no longer finite but still compact, and it preserves the orthonormality of character table.\nEach row of the table is the function e_{k}(x) of x\\in T, and the inner product between two class functions (all functions being class functions since T is abelian) is defined as with the normalizing factor |T|=1.\nThe sequence is an orthonormal basis of the space of class functions.\nFor any representation V of a finite group G, \\chi_{v} can be expressed as the span (V_{i} are the irreps of G), such that.\nSimilarly for G = T and,.\nThe Pontriagin dual \\hat{T} is and for, is its Fourier transform for.\n\n## Gelfand transform\nThe Fourier transform is also a special case of [Gelfand transform](https://bliptext.com/articles/gelfand-transform).\nIn this particular context, it is closely related to the Pontryagin duality map defined above.\nGiven an abelian [locally compact](https://bliptext.com/articles/locally-compact-space) [Hausdorff](https://bliptext.com/articles/hausdorff-space) [topological group](https://bliptext.com/articles/topological-group) G, as before we consider space\nL1(G)\n, defined using a Haar measure.\nWith convolution as multiplication,\nL1(G)\nis an abelian [Banach algebra](https://bliptext.com/articles/banach-algebra).\nIt also has an [involution](https://bliptext.com/articles/involution-mathematics) * given by\nTaking the completion with respect to the largest possibly\nC*\n-norm gives its enveloping\nC*\n-algebra, called the group\nC*\n-algebra\nC*(G)\nof G. (Any\nC*\n-norm on\nL1(G)\nis bounded by the\nL1\nnorm, therefore their supremum exists.)\nGiven any abelian\nC*\n-algebra A, the Gelfand transform gives an isomorphism between A and\nC0(A^)\n, where\nA^\nis the multiplicative linear functionals, i.e. one-dimensional representations, on A with the weak-* topology.\nThe map is simply given by\nIt turns out that the multiplicative linear functionals of\nC*(G)\n, after suitable identification, are exactly the characters of G, and the Gelfand transform, when restricted to the dense subset\nL1(G)\nis the Fourier–Pontryagin transform.\n\n## Compact non-abelian groups\nThe Fourier transform can also be defined for functions on a non-abelian group, provided that the group is [compact](https://bliptext.com/articles/compact-space).\nRemoving the assumption that the underlying group is abelian, irreducible unitary representations need not always be one-dimensional.\nThis means the Fourier transform on a non-abelian group takes values as Hilbert space operators.\nThe Fourier transform on compact groups is a major tool in [representation theory](https://bliptext.com/articles/representation-theory) and [non-commutative harmonic analysis](https://bliptext.com/articles/non-commutative-harmonic-analysis).\nLet G be a compact [Hausdorff](https://bliptext.com/articles/hausdorff-space) [topological group](https://bliptext.com/articles/topological-group).\nLet\nΣ\ndenote the collection of all isomorphism classes of finite-dimensional irreducible [unitary representations](https://bliptext.com/articles/unitary-representation), along with a definite choice of representation\nU\non the [Hilbert space](https://bliptext.com/articles/hilbert-space)\nHσ\nof finite dimension\ndσ\nfor each\nσ ∈ Σ\n. If μ is a finite [Borel measure](https://bliptext.com/articles/borel-measure) on G, then the Fourier–Stieltjes transform of μ is the operator on\nHσ\ndefined by\nwhere\nU\nis the complex-conjugate representation of\nU(σ)\nacting on\nHσ\n. If μ is [absolutely continuous](https://bliptext.com/articles/absolutely-continuous) with respect to the [left-invariant probability measure](https://bliptext.com/articles/haar-measure) λ on G, [represented](https://bliptext.com/articles/radon-nikodym-theorem) as\nfor some\nf ∈ L1(λ)\n, one identifies the Fourier transform of f with the Fourier–Stieltjes transform of μ.\nThe mapping\ndefines an isomorphism between the [Banach space](https://bliptext.com/articles/banach-space)\nM(G)\nof finite Borel measures (see [rca space](https://bliptext.com/articles/rca-space)) and a closed subspace of the Banach space\nC∞(Σ)\nconsisting of all sequences\nindexed by\nΣ\nof (bounded) linear operators\nEσ : Hσ → Hσ\nfor which the norm\nis finite.\nThe \"[convolution theorem](https://bliptext.com/articles/convolution-theorem)\" asserts that, furthermore, this isomorphism of Banach spaces is in fact an isometric isomorphism of [C*-algebras](https://bliptext.com/articles/c-algebra) into a subspace of\nC∞(Σ)\n. Multiplication on\nM(G)\nis given by [convolution](https://bliptext.com/articles/convolution) of measures and the involution * defined by\nand\nC∞(Σ)\nhas a natural\nC*\n-algebra structure as Hilbert space operators.\nThe [Peter–Weyl theorem](https://bliptext.com/articles/peter-weyl-theorem) holds, and a version of the Fourier inversion formula ([Plancherel's theorem](https://bliptext.com/articles/plancherel-s-theorem)) follows: if\nf ∈ L2(G)\n, then\nwhere the summation is understood as convergent in the\nL2\nsense.\nThe generalization of the Fourier transform to the noncommutative situation has also in part contributed to the development of [noncommutative geometry](https://bliptext.com/articles/noncommutative-geometry).\nIn this context, a categorical generalization of the Fourier transform to noncommutative groups is [Tannaka–Krein duality](https://bliptext.com/articles/tannaka-krein-duality), which replaces the group of characters with the category of representations.\nHowever, this loses the connection with harmonic functions.\n\n# Alternatives\nIn [signal processing](https://bliptext.com/articles/signal-processing) terms, a function (of time) is a representation of a signal with perfect time resolution, but no frequency information, while the Fourier transform has perfect frequency resolution, but no time information: the magnitude of the Fourier transform at a point is how much frequency content there is, but location is only given by phase (argument of the Fourier transform at a point), and [standing waves](https://bliptext.com/articles/standing-wave) are not localized in time – a sine wave continues out to infinity, without decaying.\nThis limits the usefulness of the Fourier transform for analyzing signals that are localized in time, notably [transients](https://bliptext.com/articles/transient-acoustics), or any signal of finite extent.\nAs alternatives to the Fourier transform, in [time–frequency analysis](https://bliptext.com/articles/time-frequency-analysis), one uses time–frequency transforms or time–frequency distributions to represent signals in a form that has some time information and some frequency information – by the uncertainty principle, there is a trade-off between these.\nThese can be generalizations of the Fourier transform, such as the [short-time Fourier transform](https://bliptext.com/articles/short-time-fourier-transform), [fractional Fourier transform](https://bliptext.com/articles/fractional-fourier-transform), Synchrosqueezing Fourier transform, or other functions to represent signals, as in [wavelet transforms](https://bliptext.com/articles/wavelet-transform) and [chirplet transforms](https://bliptext.com/articles/chirplet-transform), with the wavelet analog of the (continuous) Fourier transform being the [continuous wavelet transform](https://bliptext.com/articles/continuous-wavelet-transform).\n\n# Applications\nLinear operations performed in one domain (time or frequency) have corresponding operations in the other domain, which are sometimes easier to perform.\nThe operation of [differentiation](https://bliptext.com/articles/derivative) in the time domain corresponds to multiplication by the frequency, so some [differential equations](https://bliptext.com/articles/differential-equation) are easier to analyze in the frequency domain.\nAlso, [convolution](https://bliptext.com/articles/convolution) in the time domain corresponds to ordinary multiplication in the frequency domain (see [Convolution theorem](https://bliptext.com/articles/convolution-theorem)).\nAfter performing the desired operations, transformation of the result can be made back to the time domain.\n[Harmonic analysis](https://bliptext.com/articles/harmonic-analysis) is the systematic study of the relationship between the frequency and time domains, including the kinds of functions or operations that are \"simpler\" in one or the other, and has deep connections to many areas of modern mathematics.\n# [Some problems, such as certain differential equations, become easier to solve when the Fourier transform is applied. In that case the solution to the original problem is recovered using the inverse Fourier transform. | upload.wikimedia.org/wikipedia/commons/f/fd/Commutative///diagram///illustrating///problem///solving///via///the///Fourier///transform.svg]\n\n## Analysis of differential equations\nPerhaps the most important use of the Fourier transformation is to solve [partial differential equations](https://bliptext.com/articles/partial-differential-equation).\nMany of the equations of the mathematical physics of the nineteenth century can be treated this way.\nFourier studied the heat equation, which in one dimension and in dimensionless units is\nThe example we will give, a slightly more difficult one, is the wave equation in one dimension,\nAs usual, the problem is not to find a solution: there are infinitely many.\nThe problem is that of the so-called \"boundary problem\": find a solution which satisfies the \"boundary conditions\"\nHere, f and g are given functions.\nFor the heat equation, only one boundary condition can be required (usually the first one).\nBut for the wave equation, there are still infinitely many solutions y which satisfy the first boundary condition.\nBut when one imposes both conditions, there is only one possible solution.\nIt is easier to find the Fourier transform ŷ of the solution than to find the solution directly.\nThis is because the Fourier transformation takes differentiation into multiplication by the Fourier-dual variable, and so a partial differential equation applied to the original function is transformed into multiplication by polynomial functions of the dual variables applied to the transformed function.\nAfter ŷ is determined, we can apply the inverse Fourier transformation to find y.\nFourier's method is as follows.\nFirst, note that any function of the forms\nsatisfies the wave equation.\nThese are called the elementary solutions.\nSecond, note that therefore any integral\nsatisfies the wave equation for arbitrary\na+, a−, b+, b−\n. This integral may be interpreted as a continuous linear combination of solutions for the linear equation.\nNow this resembles the formula for the Fourier synthesis of a function.\nIn fact, this is the real inverse Fourier transform of\na±\nand\nb±\nin the variable x.\nThe third step is to examine how to find the specific unknown coefficient functions\na±\nand\nb±\nthat will lead to y satisfying the boundary conditions.\nWe are interested in the values of these solutions at\nt = 0\n. So we will set\nt = 0\n. Assuming that the conditions needed for Fourier inversion are satisfied, we can then find the Fourier sine and cosine transforms (in the variable x) of both sides and obtain\nand\nSimilarly, taking the derivative of y with respect to t and then applying the Fourier sine and cosine transformations yields\nand\nThese are four linear equations for the four unknowns\na±\nand\nb±\n, in terms of the Fourier sine and cosine transforms of the boundary conditions, which are easily solved by elementary algebra, provided that these transforms can be found.\nIn summary, we chose a set of elementary solutions, parametrized by ξ, of which the general solution would be a (continuous) linear combination in the form of an integral over the parameter ξ.\nBut this integral was in the form of a Fourier integral.\nThe next step was to express the boundary conditions in terms of these integrals, and set them equal to the given functions f and g. But these expressions also took the form of a Fourier integral because of the properties of the Fourier transform of a derivative.\nThe last step was to exploit Fourier inversion by applying the Fourier transformation to both sides, thus obtaining expressions for the coefficient functions\na±\nand\nb±\nin terms of the given boundary conditions f and g.\nFrom a higher point of view, Fourier's procedure can be reformulated more conceptually.\nSince there are two variables, we will use the Fourier transformation in both x and t rather than operate as Fourier did, who only transformed in the spatial variables.\nNote that ŷ must be considered in the sense of a distribution since\ny(x, t)\nis not going to be\nL1\ni2πξ\nand differentiation with respect to t to multiplication by\ni2πf\nwhere f is the frequency.\nThen the wave equation becomes an algebraic equation in ŷ:\nThis is equivalent to requiring\nŷ(ξ, f) = 0\nunless\nξ = ±f\n. Right away, this explains why the choice of elementary solutions we made earlier worked so well: obviously\nf̂ = δ(ξ ± f)\nwill be solutions.\nApplying Fourier inversion to these delta functions, we obtain the elementary solutions we picked earlier.\nBut from the higher point of view, one does not pick elementary solutions, but rather considers the space of all distributions which are supported on the (degenerate) conic\nξ − f = 0\n.\nWe may as well consider the distributions supported on the conic that are given by distributions of one variable on the line\nξ = f\nplus distributions on the line\nas follows: if Φ is any test function,\nwhere\ns+\n, and\ns−\n, are distributions of one variable.\nThen Fourier inversion gives, for the boundary conditions, something very similar to what we had more concretely above (put\nΦ(ξ, f) = ei2π(xξ+tf)\n, which is clearly of polynomial growth):\nand\nNow, as before, applying the one-variable Fourier transformation in the variable x to these functions of x yields two equations in the two unknown distributions\ns±\n(which can be taken to be ordinary functions if the boundary conditions are\nL1\nor\nL2\n).\nFrom a calculational point of view, the drawback of course is that one must first calculate the Fourier transforms of the boundary conditions, then assemble the solution from these, and then calculate an inverse Fourier transform.\nClosed form formulas are rare, except when there is some geometric symmetry that can be exploited, and the numerical calculations are difficult because of the oscillatory nature of the integrals, which makes convergence slow and hard to estimate.\nFor practical calculations, other methods are often used.\nThe twentieth century has seen the extension of these methods to all linear partial differential equations with polynomial coefficients, and by extending the notion of Fourier transformation to include Fourier integral operators, some non-linear equations as well.\n\n## Fourier-transform spectroscopy\nThe Fourier transform is also used in [nuclear magnetic resonance](https://bliptext.com/articles/nuclear-magnetic-resonance) (NMR) and in other kinds of [spectroscopy](https://bliptext.com/articles/spectroscopy), e.g. infrared ([FTIR](https://bliptext.com/articles/fourier-transform-infrared-spectroscopy)).\nIn NMR an exponentially shaped free induction decay (FID) signal is acquired in the time domain and Fourier-transformed to a Lorentzian line-shape in the frequency domain.\nThe Fourier transform is also used in [magnetic resonance imaging](https://bliptext.com/articles/magnetic-resonance-imaging) (MRI) and [mass spectrometry](https://bliptext.com/articles/mass-spectrometry).\n\n## Quantum mechanics\nThe Fourier transform is useful in [quantum mechanics](https://bliptext.com/articles/quantum-mechanics) in at least two different ways.\nTo begin with, the basic conceptual structure of quantum mechanics postulates the existence of pairs of [complementary variables](https://bliptext.com/articles/complementary-variables), connected by the [Heisenberg uncertainty principle](https://bliptext.com/articles/heisenberg-uncertainty-principle).\nFor example, in one dimension, the spatial variable q of, say, a particle, can only be measured by the quantum mechanical \"[position operator](https://bliptext.com/articles/position-operator)\" at the cost of losing information about the momentum p of the particle.\nTherefore, the physical state of the particle can either be described by a function, called \"the wave function\", of q or by a function of p but not by a function of both variables.\nThe variable p is called the conjugate variable to q. In classical mechanics, the physical state of a particle (existing in one dimension, for simplicity of exposition) would be given by assigning definite values to both p and q simultaneously.\nThus, the set of all possible physical states is the two-dimensional real vector space with a p-axis and a q-axis called the [phase space](https://bliptext.com/articles/phase-space).\nIn contrast, quantum mechanics chooses a polarisation of this space in the sense that it picks a subspace of one-half the dimension, for example, the q-axis alone, but instead of considering only points, takes the set of all complex-valued \"wave functions\" on this axis.\nNevertheless, choosing the p-axis is an equally valid polarisation, yielding a different representation of the set of possible physical states of the particle.\nBoth representations of the wavefunction are related by a Fourier transform, such that\nor, equivalently,\nPhysically realisable states are\nL2\n, and so by the [Plancherel theorem](https://bliptext.com/articles/plancherel-theorem), their Fourier transforms are also\nL2\n. (Note that since q is in units of distance and p is in units of momentum, the presence of the Planck constant in the exponent makes the exponent [dimensionless](https://bliptext.com/articles/nondimensionalization), as it should be.)\nTherefore, the Fourier transform can be used to pass from one way of representing the state of the particle, by a wave function of position, to another way of representing the state of the particle: by a wave function of momentum.\nInfinitely many different polarisations are possible, and all are equally valid.\nBeing able to transform states from one representation to another by the Fourier transform is not only convenient but also the underlying reason of the Heisenberg [uncertainty principle](https://bliptext.com/articles/).\nThe other use of the Fourier transform in both quantum mechanics and [quantum field theory](https://bliptext.com/articles/quantum-field-theory) is to solve the applicable wave equation.\nIn non-relativistic quantum mechanics, [Schrödinger's equation](https://bliptext.com/articles/schr-dinger-s-equation) for a time-varying wave function in one-dimension, not subject to external forces, is\nThis is the same as the heat equation except for the presence of the imaginary unit i. Fourier methods can be used to solve this equation.\nIn the presence of a potential, given by the potential energy function\nV(x)\n, the equation becomes\nThe \"elementary solutions\", as we referred to them above, are the so-called \"stationary states\" of the particle, and Fourier's algorithm, as described above, can still be used to solve the boundary value problem of the future evolution of ψ given its values for\n. Neither of these approaches is of much practical use in quantum mechanics.\nBoundary value problems and the time-evolution of the wave function is not of much practical interest: it is the stationary states that are most important.\nIn relativistic quantum mechanics, Schrödinger's equation becomes a wave equation as was usual in classical physics, except that complex-valued waves are considered.\nA simple example, in the absence of interactions with other particles or fields, is the free one-dimensional Klein–Gordon–Schrödinger–Fock equation, this time in dimensionless units,\nThis is, from the mathematical point of view, the same as the wave equation of classical physics solved above (but with a complex-valued wave, which makes no difference in the methods).\nThis is of great use in quantum field theory: each separate Fourier component of a wave can be treated as a separate harmonic oscillator and then quantized, a procedure known as \"second quantization\".\nFourier methods have been adapted to also deal with non-trivial interactions.\nFinally, the [number operator](https://bliptext.com/articles/quantum-harmonic-oscillator) of the [quantum harmonic oscillator](https://bliptext.com/articles/quantum-harmonic-oscillator) can be interpreted, for example via the [Mehler kernel](https://bliptext.com/articles/mehler-kernel), as the [generator](https://bliptext.com/articles/symmetry-in-quantum-mechanics) of the [Fourier transform](https://bliptext.com/articles/) \\mathcal{F}.\n\n## Signal processing\nThe Fourier transform is used for the spectral analysis of time-series.\nThe subject of statistical signal processing does not, however, usually apply the Fourier transformation to the signal itself.\nEven if a real signal is indeed transient, it has been found in practice advisable to model a signal by a function (or, alternatively, a stochastic process) which is stationary in the sense that its characteristic properties are constant over all time.\nThe Fourier transform of such a function does not exist in the usual sense, and it has been found more useful for the analysis of signals to instead take the Fourier transform of its autocorrelation function.\nThe autocorrelation function R of a function f is defined by\nThis function is a function of the time-lag τ elapsing between the values of f to be correlated.\nFor most functions f that occur in practice, R is a bounded even function of the time-lag τ and for typical noisy signals it turns out to be uniformly continuous with a maximum at\n.\nThe autocorrelation function, more properly called the autocovariance function unless it is normalized in some appropriate fashion, measures the strength of the correlation between the values of f separated by a time lag.\nThis is a way of searching for the correlation of f with its own past.\nIt is useful even for other statistical tasks besides the analysis of signals.\nFor example, if\nf(t)\nrepresents the temperature at time t, one expects a strong correlation with the temperature at a time lag of 24 hours.\nIt possesses a Fourier transform,\nThis Fourier transform is called the [power spectral density](https://bliptext.com/articles/spectral-density) function of f. (Unless all periodic components are first filtered out from f, this integral will diverge, but it is easy to filter out such periodicities.)\nThe power spectrum, as indicated by this density function P, measures the amount of variance contributed to the data by the frequency ξ.\nIn electrical signals, the variance is proportional to the average power (energy per unit time), and so the power spectrum describes how much the different frequencies contribute to the average power of the signal.\nThis process is called the spectral analysis of time-series and is analogous to the usual analysis of variance of data that is not a time-series ([ANOVA](https://bliptext.com/articles/anova)).\nKnowledge of which frequencies are \"important\" in this sense is crucial for the proper design of filters and for the proper evaluation of measuring apparatuses.\nIt can also be useful for the scientific analysis of the phenomena responsible for producing the data.\nThe power spectrum of a signal can also be approximately measured directly by measuring the average power that remains in a signal after all the frequencies outside a narrow band have been filtered out.\nSpectral analysis is carried out for visual signals as well.\nThe power spectrum ignores all phase relations, which is good enough for many purposes, but for video signals other types of spectral analysis must also be employed, still using the Fourier transform as a tool.\n\n# Other notations\nOther common notations for \\hat f(\\xi) include:\nIn the sciences and engineering it is also common to make substitutions like these:\nSo the transform pair can become\nA disadvantage of the capital letter notation is when expressing a transform such as or which become the more awkward and\nIn some contexts such as particle physics, the same symbol f may be used for both for a function as well as it Fourier transform, with the two only distinguished by their [argument](https://bliptext.com/articles/argument-of-a-function) I.e. would refer to the Fourier transform because of the momentum [argument](https://bliptext.com/articles/argument-of-a-function), while would refer to the original function because of the positional [argument](https://bliptext.com/articles/argument-of-a-function).\nAlthough tildes may be used as in \\tilde{f} to indicate Fourier transforms, tildes may also be used to indicate a modification of a quantity with a more [Lorentz invariant](https://bliptext.com/articles/lorentz-invariant) form, such as, so care must be taken.\nSimilarly, \\hat f often denotes the [Hilbert transform](https://bliptext.com/articles/hilbert-transform) of f.\nThe interpretation of the complex function\nf̂(ξ)\nmay be aided by expressing it in [polar coordinate](https://bliptext.com/articles/polar-coordinate) form\nin terms of the two real functions\nA(ξ)\nand\nφ(ξ)\nwhere:\nis the [amplitude](https://bliptext.com/articles/amplitude) and\nis the [phase](https://bliptext.com/articles/phase-waves) (see [arg function](https://bliptext.com/articles/arg-mathematics)).\nThen the inverse transform can be written:\nwhich is a recombination of all the frequency components of\nf(x)\n. Each component is a complex [sinusoid](https://bliptext.com/articles/sinusoid) of the form\ne2πixξ\nwhose amplitude is\nA(ξ)\nand whose initial [phase angle](https://bliptext.com/articles/phase-waves) (at\nx = 0\n) is\nφ(ξ)\n.\nThe Fourier transform may be thought of as a mapping on function spaces.\nThis mapping is here denoted and\n(f)\nis used to denote the Fourier transform of the function f. This mapping is linear, which means that can also be seen as a linear transformation on the function space and implies that the standard notation in linear algebra of applying a linear transformation to a vector (here the function\nf\n) can be used to write\nf\ninstead of\n(f)\n. Since the result of applying the Fourier transform is again a function, we can be interested in the value of this function evaluated at the value ξ for its variable, and this is denoted either as\nf(ξ)\nor as\n( f)(ξ)\n. Notice that in the former case, it is implicitly understood that is applied first to f and then the resulting function is evaluated at ξ, not the other way around.\nIn mathematics and various applied sciences, it is often necessary to distinguish between a function f and the value of f when its variable equals x, denoted\nf(x)\n. This means that a notation like\n(f(x))\nformally can be interpreted as the Fourier transform of the values of f at x. Despite this flaw, the previous notation appears frequently, often when a particular function or a function of a particular variable is to be transformed.\nFor example,\nis sometimes used to express that the Fourier transform of a [rectangular function](https://bliptext.com/articles/rectangular-function) is a [sinc function](https://bliptext.com/articles/sinc-function), or\nis used to express the shift property of the Fourier transform.\nNotice, that the last example is only correct under the assumption that the transformed function is a function of x, not of\nx0\n.\nAs discussed above, the [characteristic function](https://bliptext.com/articles/characteristic-function-probability-theory) of a random variable is the same as the [Fourier–Stieltjes transform](https://bliptext.com/articles/) of its distribution measure, but in this context it is typical to take a different convention for the constants.\nTypically characteristic function is defined\nAs in the case of the \"non-unitary angular frequency\" convention above, the factor of 2π appears in neither the normalizing constant nor the exponent.\nUnlike any of the conventions appearing above, this convention takes the opposite sign in the exponent.\n\n# Computation methods\nThe appropriate computation method largely depends how the original mathematical function is represented and the desired form of the output function.\nIn this section we consider both functions of a continuous variable, f(x), and functions of a discrete variable (i.e. ordered pairs of x and f values).\nFor discrete-valued x, the transform integral becomes a summation of sinusoids, which is still a continuous function of frequency (\\xi or \\omega).\nWhen the sinusoids are harmonically-related (i.e. when the x-values are spaced at integer multiples of an interval), the transform is called [discrete-time Fourier transform](https://bliptext.com/articles/discrete-time-fourier-transform) (DTFT).\n\n## Discrete Fourier transforms and fast Fourier transforms\nSampling the DTFT at equally-spaced values of frequency is the most common modern method of computation.\nEfficient procedures, depending on the frequency resolution needed, are described at.\nThe [discrete Fourier transform](https://bliptext.com/articles/discrete-fourier-transform) (DFT), used there, is usually computed by a [fast Fourier transform](https://bliptext.com/articles/fast-fourier-transform) (FFT) algorithm.\n\n## Analytic integration of closed-form functions\nTables of [closed-form](https://bliptext.com/articles/closed-form-expression) Fourier transforms, such as and, are created by mathematically evaluating the Fourier analysis integral (or summation) into another [closed-form](https://bliptext.com/articles/closed-form-expression) function of frequency (\\xi or \\omega).\nWhen mathematically possible, this provides a transform for a continuum of frequency values.\nMany computer algebra systems such as [Matlab](https://bliptext.com/articles/matlab) and [Mathematica](https://bliptext.com/articles/mathematica) that are capable of [symbolic integration](https://bliptext.com/articles/symbolic-integration) are capable of computing Fourier transforms analytically.\nFor example, to compute the Fourier transform of\ncos(6πt) e−πt 2\none might enter the command integrate cos(6*pi*t) exp(−pi*t^2) exp(-i*2*pi*f*t) from -inf to inf into [Wolfram Alpha](https://bliptext.com/articles/wolfram-alpha).\n\n## Numerical integration of closed-form continuous functions\nDiscrete sampling of the Fourier transform can also be done by [numerical integration](https://bliptext.com/articles/numerical-integration) of the definition at each value of frequency for which transform is desired.\nThe numerical integration approach works on a much broader class of functions than the analytic approach.\n\n## Numerical integration of a series of ordered pairs\nIf the input function is a series of ordered pairs, numerical integration reduces to just a summation over the set of data pairs.\nThe DTFT is a common subcase of this more general situation.\n\n# Tables of important Fourier transforms\nThe following tables record some closed-form Fourier transforms.\nFor functions\nf(x)\nand\ng(x)\ndenote their Fourier transforms by\nf̂\nand\nĝ\n. Only the three most common conventions are included.\nIt may be useful to notice that entry 105 gives a relationship between the Fourier transform of a function and the original function, which can be seen as relating the Fourier transform and its inverse.\n\n## Functional relationships, one-dimensional\nThe Fourier transforms in this table may be found in or.\n\n## Square-integrable functions, one-dimensional\nThe Fourier transforms in this table may be found in, , or.\n\n## Distributions, one-dimensional\nThe Fourier transforms in this table may be found in or.\n\n## Two-dimensional functions\n\n## Formulas for general \n\nn\n\n-dimensional functions\n\n# Citations","wordIndex":5,"createdAt":"2025-03-08T02:32:07.480Z","user":{"id":"E6iXdKCEcr2dpx7wlFc1v38vYpQIgHKW","name":"DynamicArrays","image":"https://cdn.discordapp.com/avatars/887802393054572584/bd9d90fedebd723bbc46de9db3ffba71.png"}}],"article":{"id":"d575f5e9-93a0-4485-b993-aedcdb18861b","title":"Fourier transform","slug":"fourier-transform","content":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), the SigmaGyatt (**FT**) is an [integral transform](https://bliptext.com/articles/integral-transform) that takes a [function](https://bliptext.com/articles/function-mathematics) as input and outputs another [function](https://bliptext.com/articles/function-mathematics) that describes the extent to which various frequencies are present in the original [function](https://bliptext.com/articles/function-mathematics).\nThe output of the transform is a [complex](https://bliptext.com/articles/complex-number)-valued function of frequency.\nThe term Fourier transform refers to both this complex-valued function and the [mathematical operation](https://bliptext.com/articles/operation-mathematics).\nWhen a distinction needs to be made, the output of the operation is sometimes called the [frequency domain](https://bliptext.com/articles/frequency-domain) representation of the original function.\nThe Fourier transform is analogous to decomposing the [sound](https://bliptext.com/articles/sound) of a musical [chord](https://bliptext.com/articles/chord-music) into the [intensities](https://bliptext.com/articles/sound-intensity) of its constituent [pitches](https://bliptext.com/articles/pitch-music).\nFunctions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa, a phenomenon known as the [uncertainty principle](https://bliptext.com/articles/).\nThe [critical](https://bliptext.com/articles/critical-point-mathematics) case for this principle is the [Gaussian function](https://bliptext.com/articles/gaussian-function), of substantial importance in [probability theory](https://bliptext.com/articles/probability-theory) and [statistics](https://bliptext.com/articles/statistics) as well as in the study of physical phenomena exhibiting [normal distribution](https://bliptext.com/articles/normal-distribution) (e.g., [diffusion](https://bliptext.com/articles/diffusion)).\nThe Fourier transform of a Gaussian function is another Gaussian function.\n[Joseph Fourier](https://bliptext.com/articles/joseph-fourier) introduced [sine and cosine transforms](https://bliptext.com/articles/sine-and-cosine-transforms) (which [correspond to the imaginary and real components](https://bliptext.com/articles/sine-and-cosine-transforms) of the modern Fourier transform) in his study of [heat transfer](https://bliptext.com/articles/heat-transfer), where Gaussian functions appear as solutions of the [heat equation](https://bliptext.com/articles/heat-equation).\nThe Fourier transform can be formally defined as an [improper](https://bliptext.com/articles/improper-integral) [Riemann integral](https://bliptext.com/articles/riemann-integral), making it an integral transform, although this definition is not suitable for many applications requiring a more sophisticated integration theory.\nFor example, many relatively simple applications use the [Dirac delta function](https://bliptext.com/articles/dirac-delta-function), which can be treated formally as if it were a function, but the justification requires a mathematically more sophisticated viewpoint.\nThe Fourier transform can also be generalized to functions of several variables on [Euclidean space](https://bliptext.com/articles/euclidean-space), sending a function of 3-dimensional 'position space' to a function of 3-dimensional momentum (or a function of space and time to a function of [4-momentum](https://bliptext.com/articles/4-momentum)).\nThis idea makes the spatial Fourier transform very natural in the study of waves, as well as in [quantum mechanics](https://bliptext.com/articles/quantum-mechanics), where it is important to be able to represent wave solutions as functions of either position or momentum and sometimes both.\nIn general, functions to which Fourier methods are applicable are complex-valued, and possibly [vector-valued](https://bliptext.com/articles/vector-valued-function).\nStill further generalization is possible to functions on [groups](https://bliptext.com/articles/group-mathematics), which, besides the original Fourier transform on [[Real number#Arithmetic|\nR\n]] or\nRn\n, notably includes the [discrete-time Fourier transform](https://bliptext.com/articles/discrete-time-fourier-transform) (DTFT, group =\nZ\n), the [discrete Fourier transform](https://bliptext.com/articles/discrete-fourier-transform) (DFT, group = [[cyclic group|\nZ mod N\n]]) and the [Fourier series](https://bliptext.com/articles/fourier-series) or circular Fourier transform (group =\nS1\n, the unit circle ≈ closed finite interval with endpoints identified). The latter is routinely employed to handle [periodic functions](https://bliptext.com/articles/periodic-function). The [fast Fourier transform](https://bliptext.com/articles/fast-fourier-transform) (FFT) is an algorithm for computing the DFT.\n# [An example application of the Fourier transform is determining the constituent pitches in a musical waveform. This image is the result of applying a constant-Q transform (a Fourier-related transform) to the waveform of a C major piano chord. The first three peaks on the left correspond to the frequencies of the fundamental frequency of the chord (C, E, G). The remaining smaller peaks are higher-frequency overtones of the fundamental pitches. A pitch detection algorithm could use the relative intensity of these peaks to infer which notes the pianist pressed. | upload.wikimedia.org/wikipedia/commons/3/31/CQT-piano-chord.png]\n\n# Definition\nThe Fourier transform is an analysis process, decomposing a complex-valued function into its constituent frequencies and their amplitudes.\nThe inverse process is synthesis, which recreates from its transform.\nWe can start with an analogy, the **[Fourier series](https://bliptext.com/articles/fourier-series)**, which analyzes on a bounded interval for some positive real number P. The constituent frequencies are a discrete set of harmonics at frequencies whose amplitude and phase are given by the **analysis formula****:******The actual **[Fourier series](https://bliptext.com/articles/fourier-series)** is the **synthesis formula****:******On an unbounded interval, P\\to\\infty, the constituent frequencies are a continuum****:**** and c_n is replaced by a function****:****\nEvaluating for all values of \\xi produces the frequency-domain function.\nThe integral can diverge at some frequencies.\n(see ) But it converges for ** all ** frequencies when f(x) decays with** all **derivatives as **:**.\n(See [Schwartz function](https://bliptext.com/articles/schwartz-function)).\nBy the [Riemann–Lebesgue lemma](https://bliptext.com/articles/riemann-lebesgue-lemma), the transformed function \\widehat f also decays with all derivatives.\nThe complex number, in polar coordinates, conveys both [amplitude](https://bliptext.com/articles/amplitude) and [phase](https://bliptext.com/articles/phase-offset) of frequency \\xi.\nThe intuitive interpretation of is that the effect of multiplying f(x) by is to subtract \\xi from every frequency component of function f(x).\nOnly the component that was at frequency \\xi can produce a non-zero value of the infinite integral, because (at least formally) all the other shifted components are oscillatory and integrate to zero.\n(see )\nThe corresponding synthesis formula is:\nis a representation of f(x) as a weighted summation of complex exponential functions.\nThis is also known as the [Fourier inversion theorem](https://bliptext.com/articles/fourier-inversion-theorem), and was first introduced in [Fourier's](https://bliptext.com/articles/joseph-fourier) Analytical Theory of Heat.\nThe functions f and \\widehat{f} are referred to as a **Fourier transform pair**.\nA common notation for designating transform pairs is**:**\nfor example\n\n## Definition for Lebesgue integrable functions\nUntil now, we have been dealing with Schwartz functions, which decay rapidly at infinity, with all derivatives.\nThis excludes many functions of practical importance from the definition, such as the [rect function](https://bliptext.com/articles/rect-function).\nA [measurable function](https://bliptext.com/articles/measurable-function) is called (Lebesgue) integrable if the [Lebesgue integral](https://bliptext.com/articles/lebesgue-integral) of its absolute value is finite:\nTwo measurable functions are equivalent if they are equal except on a set of measure zero.\nThe set of all equivalence classes of integrable functions is denoted.\nThen:\nThe integral is well-defined for all because of the assumption.\n(It can be shown that the function is bounded and [uniformly continuous](https://bliptext.com/articles/uniformly-continuous) in the frequency domain, and moreover, by the [Riemann–Lebesgue lemma](https://bliptext.com/articles/riemann-lebesgue-lemma), it is zero at infinity.)\nHowever, the class of Lebesgue integrable functions is not ideal from the point of view of the Fourier transform because there is no easy characterization of the image, and thus no easy characterization of the inverse transform.\n\n## Unitarity and definition for square integrable functions\nWhile defines the Fourier transform for (complex-valued) functions in, it is easy to see that it is not well-defined for other integrability classes, most importantly.\nFor functions in, and with the conventions of , the Fourier transform is a [unitary operator](https://bliptext.com/articles/unitary-operator) with respect to the Hilbert inner product on , restricted to the dense subspace of integrable functions.\nTherefore, it admits a unique continuous extension to a unitary operator on, also called the Fourier transform.\nThis extension is important in part because the Fourier transform preserves the space so that, unlike the case of L^1, the Fourier transform and inverse transform are on the same footing, being transformations of the same space of functions to itself.\nImportantly, for functions in L^2, the Fourier transform is no longer given by (interpreted as a Lebesgue integral).\nFor example, the function is in L^2 but not L^1, so the integral diverges.\nIn such cases, the Fourier transform can be obtained explicitly by regularizing the integral, and then passing to a limit.\nIn practice, the integral is often regarded as an [improper integral](https://bliptext.com/articles/improper-integral) instead of a proper Lebesgue integral, but sometimes for convergence one needs to use [weak limit](https://bliptext.com/articles/weak-limit) or [principal value](https://bliptext.com/articles/cauchy-principal-value) instead of the (pointwise) limits implicit in an [improper integral](https://bliptext.com/articles/improper-integral).\nand each gives three rigorous ways of extending the Fourier transform to square integrable functions using this procedure.\nThe conventions chosen in this article are those of [harmonic analysis](https://bliptext.com/articles/harmonic-analysis), and are characterized as the unique conventions such that the Fourier transform is both [unitary](https://bliptext.com/articles/unitary-operator) on\nL2\nand an algebra homomorphism from\nL1\nto\nL∞\n, without renormalizing the Lebesgue measure.\n\n## Angular frequency (ω)\nWhen the independent variable (x) represents time (often denoted by t), the transform variable (\\xi) represents [frequency](https://bliptext.com/articles/frequency) (often denoted by f).\nFor example, if time is measured in [seconds](https://bliptext.com/articles/second), then frequency is in [hertz](https://bliptext.com/articles/hertz).\nThe Fourier transform can also be written in terms of [angular frequency](https://bliptext.com/articles/angular-frequency), whose units are [radians](https://bliptext.com/articles/radian) per second.\nThe substitution into produces this convention, where function \\widehat f is relabeled\nUnlike the definition, the Fourier transform is no longer a [unitary transformation](https://bliptext.com/articles/unitary-transformation), and there is less symmetry between the formulas for the transform and its inverse.\nThose properties are restored by splitting the 2 \\pi factor evenly between the transform and its inverse, which leads to another convention:\nVariations of all three conventions can be created by conjugating the complex-exponential [kernel](https://bliptext.com/articles/integral-kernel) of both the forward and the reverse transform.\nThe signs must be opposites.\n\n## Extension of the definition\nFor 1 0\nsuch that for any [integer](https://bliptext.com/articles/integer)\nn ≥ 0\n,\nfor some constant C. (In this case, f is supported on\n[−a, a]\n.) This can be expressed by saying that\nf̂\nis an [entire function](https://bliptext.com/articles/entire-function) which is [rapidly decreasing](https://bliptext.com/articles/rapidly-decreasing) in σ (for fixed τ) and of exponential growth in τ (uniformly in σ).\n(If f is not smooth, but only\nL2\n, the statement still holds provided\n. ) The space of such functions of a [complex variable](https://bliptext.com/articles/complex-analysis) is called the Paley—Wiener space. This theorem has been generalised to semisimple [Lie groups](https://bliptext.com/articles/lie-group).\nIf f is supported on the half-line\nt ≥ 0\n, then f is said to be \"causal\" because the [impulse response function](https://bliptext.com/articles/impulse-response-function) of a physically realisable [filter](https://bliptext.com/articles/filter-mathematics) must have this property, as no effect can precede its cause.\n[Paley](https://bliptext.com/articles/raymond-paley) and Wiener showed that then\nf̂\nextends to a [holomorphic function](https://bliptext.com/articles/holomorphic-function) on the complex lower half-plane\nτ < 0\nwhich tends to zero as τ goes to infinity.\nThe converse is false and it is not known how to characterise the Fourier transform of a causal function.\n\n## Laplace transform\nThe Fourier transform\nf̂(ξ)\nis related to the [Laplace transform](https://bliptext.com/articles/laplace-transform)\nF(s)\n, which is also used for the solution of [differential equations](https://bliptext.com/articles/differential-equation) and the analysis of [filters](https://bliptext.com/articles/filter-signal-processing).\nIt may happen that a function f for which the Fourier integral does not converge on the real axis at all, nevertheless has a complex Fourier transform defined in some region of the [complex plane](https://bliptext.com/articles/complex-plane).\nFor example, if\nf(t)\nis of exponential growth, i.e.,\nfor some constants\nC, a ≥ 0\n, then\nconvergent for all\n2πτ < −a\n, is the [two-sided Laplace transform](https://bliptext.com/articles/two-sided-laplace-transform) of f.\nThe more usual version (\"one-sided\") of the Laplace transform is\nIf f is also causal, and analytical, then: Thus, extending the Fourier transform to the complex domain means it includes the Laplace transform as a special case in the case of causal functions—but with the change of variable\n.\nFrom another, perhaps more classical viewpoint, the Laplace transform by its form involves an additional exponential regulating term which lets it converge outside of the imaginary line where the Fourier transform is defined.\nAs such it can converge for at most exponentially divergent series and integrals, whereas the original Fourier decomposition cannot, enabling analysis of systems with divergent or critical elements.\nTwo particular examples from linear signal processing are the construction of allpass filter networks from critical comb and mitigating filters via exact pole-zero cancellation on the unit circle.\nSuch designs are common in audio processing, where highly nonlinear phase response is sought for, as in reverb.\nFurthermore, when extended pulselike impulse responses are sought for signal processing work, the easiest way to produce them is to have one circuit which produces a divergent time response, and then to cancel its divergence through a delayed opposite and compensatory response.\nThere, only the delay circuit in-between admits a classical Fourier description, which is critical.\nBoth the circuits to the side are unstable, and do not admit a convergent Fourier decomposition.\nHowever, they do admit a Laplace domain description, with identical half-planes of convergence in the complex plane (or in the discrete case, the Z-plane), wherein their effects cancel.\nIn modern mathematics the Laplace transform is conventionally subsumed under the aegis Fourier methods.\nBoth of them are subsumed by the far more general, and more abstract, idea of [harmonic analysis](https://bliptext.com/articles/harmonic-analysis).\n\n## Inversion\nStill with, if \\widehat f is complex analytic for\na ≤ τ ≤ b\n, then\nby [Cauchy's integral theorem](https://bliptext.com/articles/cauchy-s-integral-theorem).\nTherefore, the Fourier inversion formula can use integration along different lines, parallel to the real axis.\nTheorem: If\nf(t) = 0\nfor\nt < 0\n, and\nfor some constants\nC, a > 0\n, then\nfor any\nτ < −a⁄2π\n.\nThis theorem implies the [Mellin inversion formula](https://bliptext.com/articles/inverse-laplace-transform) for the Laplace transformation,\nfor any\nb > a\n, where\nF(s)\nis the Laplace transform of\nf(t)\n.\nThe hypotheses can be weakened, as in the results of Carleson and Hunt, to\nf(t) e−at\nbeing\nL1\n, provided that f be of bounded variation in a closed neighborhood of t (cf. [Dini test](https://bliptext.com/articles/dini-test)), the value of f at t be taken to be the [arithmetic mean](https://bliptext.com/articles/arithmetic-mean) of the left and right limits, and that the integrals be taken in the sense of Cauchy principal values.\nL2\nversions of these inversion formulas are also available.\n\n# Fourier transform on Euclidean space\nThe Fourier transform can be defined in any arbitrary number of dimensions n. As with the one-dimensional case, there are many conventions.\nFor an integrable function\nf(x)\n, this article takes the definition:\nwhere\nx\nand\nξ\nare n-dimensional [vectors](https://bliptext.com/articles/vector-mathematics), and\nx · ξ\nis the [dot product](https://bliptext.com/articles/dot-product) of the vectors.\nAlternatively,\nξ\ncan be viewed as belonging to the [dual vector space](https://bliptext.com/articles/dual-space) \\R^{n\\star}, in which case the dot product becomes the [contraction](https://bliptext.com/articles/tensor-contraction) of\nx\nand\nξ\n, usually written as\n⟨x, ξ⟩\n.\nAll of the basic properties listed above hold for the n-dimensional Fourier transform, as do Plancherel's and Parseval's theorem.\nWhen the function is integrable, the Fourier transform is still uniformly continuous and the [Riemann–Lebesgue lemma](https://bliptext.com/articles/riemann-lebesgue-lemma) holds.\n\n## Uncertainty principle\nGenerally speaking, the more concentrated\nf(x)\nis, the more spread out its Fourier transform\nf̂(ξ)\nmust be.\nIn particular, the scaling property of the Fourier transform may be seen as saying: if we squeeze a function in x, its Fourier transform stretches out in ξ.\nIt is not possible to arbitrarily concentrate both a function and its Fourier transform.\nThe trade-off between the compaction of a function and its Fourier transform can be formalized in the form of an [uncertainty principle](https://bliptext.com/articles/uncertainty-principle) by viewing a function and its Fourier transform as [conjugate variables](https://bliptext.com/articles/conjugate-variables) with respect to the [symplectic form](https://bliptext.com/articles/symplectic-form) on the [time–frequency domain](https://bliptext.com/articles/time-frequency-representation): from the point of view of the [linear canonical transformation](https://bliptext.com/articles/linear-canonical-transformation), the Fourier transform is rotation by 90° in the [time–frequency domain](https://bliptext.com/articles/time-frequency-representation), and preserves the [symplectic form](https://bliptext.com/articles/symplectic-form).\nSuppose\nf(x)\nis an integrable and [square-integrable](https://bliptext.com/articles/square-integrable) function.\nWithout loss of generality, assume that\nf(x)\nis normalized:\nIt follows from the [Plancherel theorem](https://bliptext.com/articles/plancherel-theorem) that\nf̂(ξ)\nis also normalized.\nThe spread around\nmay be measured by the dispersion about zero defined by\nIn probability terms, this is the [second moment](https://bliptext.com/articles/moment-mathematics) of\nabout zero.\nThe uncertainty principle states that, if\nf(x)\nis absolutely continuous and the functions\nx·f(x)\nand\nf(x)\nare square integrable, then\nThe equality is attained only in the case\nwhere\nσ > 0\nis arbitrary and\nC1 = 4√2⁄√σ\nso that f is\nL2\n-normalized.\nIn other words, where f is a (normalized) [Gaussian function](https://bliptext.com/articles/gaussian-function) with variance\nσ2/2π\n, centered at zero, and its Fourier transform is a Gaussian function with variance\nσ−2/2π\n.\nIn fact, this inequality implies that:\nfor any\nx0\n,\nξ0 ∈ R\n.\nIn [quantum mechanics](https://bliptext.com/articles/quantum-mechanics), the [momentum](https://bliptext.com/articles/momentum) and position [wave functions](https://bliptext.com/articles/wave-function) are Fourier transform pairs, up to a factor of the [Planck constant](https://bliptext.com/articles/planck-constant).\nWith this constant properly taken into account, the inequality above becomes the statement of the [Heisenberg uncertainty principle](https://bliptext.com/articles/heisenberg-uncertainty-principle).\nA stronger uncertainty principle is the [Hirschman uncertainty principle](https://bliptext.com/articles/hirschman-uncertainty), which is expressed as:\nwhere\nH(p)\nis the [differential entropy](https://bliptext.com/articles/differential-entropy) of the [probability density function](https://bliptext.com/articles/probability-density-function)\np(x)\nwhere the logarithms may be in any base that is consistent.\nThe equality is attained for a Gaussian, as in the previous case.\n\n## Sine and cosine transforms\nFourier's original formulation of the transform did not use complex numbers, but rather sines and cosines.\nStatisticians and others still use this form.\nAn absolutely integrable function f for which Fourier inversion holds can be expanded in terms of genuine frequencies (avoiding negative frequencies, which are sometimes considered hard to interpret physically ) λ by\nThis is called an expansion as a trigonometric integral, or a Fourier integral expansion.\nThe coefficient functions a and b can be found by using variants of the Fourier cosine transform and the Fourier sine transform (the normalisations are, again, not standardised):\nand\nOlder literature refers to the two transform functions, the Fourier cosine transform, a, and the Fourier sine transform, b.\nThe function f can be recovered from the sine and cosine transform using\ntogether with trigonometric identities.\nThis is referred to as Fourier's integral formula.\n\n## Spherical harmonics\nLet the set of [homogeneous](https://bliptext.com/articles/homogeneous-polynomial) [harmonic](https://bliptext.com/articles/harmonic-function) [polynomials](https://bliptext.com/articles/polynomial) of degree k on\nRn\nbe denoted by\nAk\n. The set\nAk\nconsists of the [solid spherical harmonics](https://bliptext.com/articles/solid-spherical-harmonics) of degree k. The [solid spherical harmonics](https://bliptext.com/articles/solid-spherical-harmonics) play a similar role in higher dimensions to the Hermite polynomials in dimension one.\nSpecifically, if\nf(x) = e−π P(x)\nfor some\nP(x)\nin\nAk\n, then\nf̂(ξ) = i f(ξ)\n. Let the set\nHk\nbe the closure in\nL2(Rn)\nof linear combinations of functions of the form\nf(\nwhere\nP(x)\nis in\nAk\n. The space\nL2(Rn)\nis then a direct sum of the spaces\nHk\nand the Fourier transform maps each space\nHk\nto itself and is possible to characterize the action of the Fourier transform on each space\nHk\n.\nLet\n)P(x)\n(with\nP(x)\nin\nAk\n), then\nwhere\nHere\nJ(n + 2k − 2)/2\ndenotes the [Bessel function](https://bliptext.com/articles/bessel-function) of the first kind with order\nn + 2k − 2⁄2\n. When\nthis gives a useful formula for the Fourier transform of a radial function.\nThis is essentially the [Hankel transform](https://bliptext.com/articles/hankel-transform).\nMoreover, there is a simple recursion relating the cases\nn + 2\nand n allowing to compute, e.g., the three-dimensional Fourier transform of a radial function from the one-dimensional one.\n\n## Restriction problems\nIn higher dimensions it becomes interesting to study restriction problems for the Fourier transform.\nThe Fourier transform of an integrable function is continuous and the restriction of this function to any set is defined.\nBut for a square-integrable function the Fourier transform could be a general class of square integrable functions.\nAs such, the restriction of the Fourier transform of an\nL2(Rn)\nfunction cannot be defined on sets of measure 0.\nIt is still an active area of study to understand restriction problems in\nL\nfor\n1 < p < 2\n. It is possible in some cases to define the restriction of a Fourier transform to a set S, provided S has non-zero curvature.\nThe case when S is the unit sphere in\nRn\nis of particular interest.\nIn this case the Tomas–[Stein](https://bliptext.com/articles/elias-stein) restriction theorem states that the restriction of the Fourier transform to the unit sphere in\nRn\nis a bounded operator on\nL\nprovided\n1 ≤ p ≤ 2n + 2⁄n + 3\n.\nOne notable difference between the Fourier transform in 1 dimension versus higher dimensions concerns the partial sum operator.\nConsider an increasing collection of measurable sets\nER\nindexed by\nR ∈ (0,∞)\n2R\n. For a given integrable function f, consider the function fR defined by:\nSuppose in addition that\nf ∈ L(Rn)\n. For\nand\n1 < p < ∞\n, if one takes\n, then fR converges to f in\nL\nas R tends to infinity, by the boundedness of the [Hilbert transform](https://bliptext.com/articles/hilbert-transform).\nNaively one may hope the same holds true for\nn > 1\n. In the case that ER is taken to be a cube with side length R, then convergence still holds.\nAnother natural candidate is the Euclidean ball {{math|E{{sub|R}} = {ξ : {{abs|ξ}} < R{{)}}}}. In order for this partial sum operator to converge, it is necessary that the multiplier for the unit ball be bounded in\nL(Rn)\n. For\nn ≥ 2\nit is a celebrated theorem of [Charles Fefferman](https://bliptext.com/articles/charles-fefferman) that the multiplier for the unit ball is never bounded unless {{math|p {{=}} 2}}.\nIn fact, when\np ≠ 2\n, this shows that not only may fR fail to converge to f in\nL\n, but for some functions\nf ∈ L(Rn)\n, fR is not even an element of\nL\n.\n\n# Fourier transform on function spaces\n\n## On Lp spaces\n\n### On L1\nThe definition of the Fourier transform by the integral formula\nis valid for Lebesgue integrable functions f; that is,\nf ∈ L1(Rn)\n.\nThe Fourier transform\nis a [bounded operator](https://bliptext.com/articles/bounded-operator).\nThis follows from the observation that\nwhich shows that its [operator norm](https://bliptext.com/articles/operator-norm) is bounded by 1.\nIndeed, it equals 1, which can be seen, for example, from the [transform of the rect function](https://bliptext.com/articles/).\nThe image of\nL1\nis a subset of the space\nC0(Rn)\nof continuous functions that tend to zero at infinity (the [Riemann–Lebesgue lemma](https://bliptext.com/articles/riemann-lebesgue-lemma)), although it is not the entire space.\nIndeed, there is no simple characterization of the image.\n\n### On L2\nSince compactly supported smooth functions are integrable and dense in\nL2(Rn)\n, the [Plancherel theorem](https://bliptext.com/articles/plancherel-theorem) allows one to extend the definition of the Fourier transform to general functions in\nL2(Rn)\nby continuity arguments.\nThe Fourier transform in\nL2(Rn)\nis no longer given by an ordinary Lebesgue integral, although it can be computed by an [improper integral](https://bliptext.com/articles/improper-integral), here meaning that for an\nL2\nfunction f,\nwhere the limit is taken in the\nL2\nsense.\n)\nMany of the properties of the Fourier transform in\nL1\ncarry over to\nL2\n, by a suitable limiting argument.\nFurthermore,\nis a [unitary operator](https://bliptext.com/articles/unitary-operator).\nFor an operator to be unitary it is sufficient to show that it is bijective and preserves the inner product, so in this case these follow from the Fourier inversion theorem combined with the fact that for any\nf, g ∈ L2(Rn)\nwe have\nIn particular, the image of\nL2(Rn)\nis itself under the Fourier transform.\n\n### On other Lp\nThe definition of the Fourier transform can be extended to functions in\nL(Rn)\nfor\n1 ≤ p ≤ 2\nby decomposing such functions into a fat tail part in\nL2\nplus a fat body part in\nL1\n. In each of these spaces, the Fourier transform of a function in\nL(Rn)\nis in\nL(Rn)\n, where\nq = p⁄p − 1\nis the [Hölder conjugate](https://bliptext.com/articles/h-lder-conjugate) of p (by the [Hausdorff–Young inequality](https://bliptext.com/articles/hausdorff-young-inequality)).\nHowever, except for\np = 2\n, the image is not easily characterized.\nFurther extensions become more technical.\nThe Fourier transform of functions in\nL\nfor the range\n2 < p < ∞\nrequires the study of distributions.\nIn fact, it can be shown that there are functions in\nL\nwith\np > 2\nso that the Fourier transform is not defined as a function.\n\n## Tempered distributions\nOne might consider enlarging the domain of the Fourier transform from\nL1 + L2\nby considering [generalized functions](https://bliptext.com/articles/generalized-function), or distributions.\nA distribution on\nRn\nis a continuous linear functional on the space\nCc(Rn)\nof compactly supported smooth functions, equipped with a suitable topology.\nThe strategy is then to consider the action of the Fourier transform on\nCc(Rn)\nand pass to distributions by duality.\nThe obstruction to doing this is that the Fourier transform does not map\nCc(Rn)\nto\nCc(Rn)\n. In fact the Fourier transform of an element in\nCc(Rn)\ncan not vanish on an open set; see the above discussion on the uncertainty principle.\nThe right space here is the slightly larger space of [Schwartz functions](https://bliptext.com/articles/schwartz-space).\nThe Fourier transform is an automorphism on the Schwartz space, as a topological vector space, and thus induces an automorphism on its dual, the space of tempered distributions.\nThe tempered distributions include all the integrable functions mentioned above, as well as well-behaved functions of polynomial growth and distributions of compact support.\nFor the definition of the Fourier transform of a tempered distribution, let f and g be integrable functions, and let\nf̂\nand\nĝ\nbe their Fourier transforms respectively.\nThen the Fourier transform obeys the following multiplication formula,\nEvery integrable function f defines (induces) a distribution Tf by the relation\nfor all Schwartz functions φ.\nSo it makes sense to define Fourier transform\nT̂f\nof\nTf\nby\nfor all Schwartz functions φ.\nExtending this to all tempered distributions T gives the general definition of the Fourier transform.\nDistributions can be differentiated and the above-mentioned compatibility of the Fourier transform with differentiation and convolution remains true for tempered distributions.\n\n# Generalizations\n\n## Fourier–Stieltjes transform\nThe Fourier transform of a [finite](https://bliptext.com/articles/finite-measure) [Borel measure](https://bliptext.com/articles/borel-measure) μ on\nRn\nis given by:\nThis transform continues to enjoy many of the properties of the Fourier transform of integrable functions.\nOne notable difference is that the [Riemann–Lebesgue lemma](https://bliptext.com/articles/riemann-lebesgue-lemma) fails for measures.\nIn the case that\n, then the formula above reduces to the usual definition for the Fourier transform of f. In the case that μ is the probability distribution associated to a random variable X, the Fourier–Stieltjes transform is closely related to the [characteristic function](https://bliptext.com/articles/characteristic-function-probability-theory), but the typical conventions in probability theory take\neiξx\ninstead of\ne−i2πξx\n. In the case when the distribution has a [probability density function](https://bliptext.com/articles/probability-density-function) this definition reduces to the Fourier transform applied to the [probability density function](https://bliptext.com/articles/probability-density-function), again with a different choice of constants.\nThe Fourier transform may be used to give a characterization of measures.\n[Bochner's theorem](https://bliptext.com/articles/bochner-s-theorem) characterizes which functions may arise as the Fourier–Stieltjes transform of a positive measure on the circle.\nFurthermore, the [Dirac delta function](https://bliptext.com/articles/dirac-delta-function), although not a function, is a finite Borel measure.\nIts Fourier transform is a constant function (whose specific value depends upon the form of the Fourier transform used).\n\n## Locally compact abelian groups\nThe Fourier transform may be generalized to any locally compact abelian group.\nA [locally compact](https://bliptext.com/articles/locally-compact) [abelian group](https://bliptext.com/articles/abelian-group) is an [abelian group](https://bliptext.com/articles/abelian-group) that is at the same time a [locally compact](https://bliptext.com/articles/locally-compact) [Hausdorff topological space](https://bliptext.com/articles/hausdorff-space) so that the group operation is continuous.\nIf G is a locally compact abelian group, it has a translation invariant measure μ, called [Haar measure](https://bliptext.com/articles/haar-measure).\nFor a locally compact abelian group G, the set of irreducible, i.e. one-dimensional, unitary representations are called its [characters](https://bliptext.com/articles/character-group).\nWith its natural group structure and the topology of uniform convergence on compact sets (that is, the topology induced by the [compact-open topology](https://bliptext.com/articles/compact-open-topology) on the space of all continuous functions from G to the [circle group](https://bliptext.com/articles/circle-group)), the set of characters Ĝ is itself a locally compact abelian group, called the Pontryagin dual of G. For a function f in\nL1(G)\n, its Fourier transform is defined by\nThe Riemann–Lebesgue lemma holds in this case;\nf̂(ξ)\nis a function vanishing at infinity on Ĝ.\nThe Fourier transform on is an example; here T is a locally compact abelian group, and the Haar measure μ on T can be thought of as the Lebesgue measure on [0,1). Consider the representation of T on the complex plane C that is a 1-dimensional complex vector space. There are a group of representations (which are irreducible since C is 1-dim) where for x\\in T.\nThe character of such representation, that is the trace of e_{k}(x) for each x\\in T and k\\in Z, is itself.\nIn the case of representation of finite group, the character table of the group G are rows of vectors such that each row is the character of one irreducible representation of G, and these vectors form an orthonormal basis of the space of class functions that map from G to C by Schur's lemma.\nNow the group T is no longer finite but still compact, and it preserves the orthonormality of character table.\nEach row of the table is the function e_{k}(x) of x\\in T, and the inner product between two class functions (all functions being class functions since T is abelian) is defined as with the normalizing factor |T|=1.\nThe sequence is an orthonormal basis of the space of class functions.\nFor any representation V of a finite group G, \\chi_{v} can be expressed as the span (V_{i} are the irreps of G), such that.\nSimilarly for G = T and,.\nThe Pontriagin dual \\hat{T} is and for, is its Fourier transform for.\n\n## Gelfand transform\nThe Fourier transform is also a special case of [Gelfand transform](https://bliptext.com/articles/gelfand-transform).\nIn this particular context, it is closely related to the Pontryagin duality map defined above.\nGiven an abelian [locally compact](https://bliptext.com/articles/locally-compact-space) [Hausdorff](https://bliptext.com/articles/hausdorff-space) [topological group](https://bliptext.com/articles/topological-group) G, as before we consider space\nL1(G)\n, defined using a Haar measure.\nWith convolution as multiplication,\nL1(G)\nis an abelian [Banach algebra](https://bliptext.com/articles/banach-algebra).\nIt also has an [involution](https://bliptext.com/articles/involution-mathematics) * given by\nTaking the completion with respect to the largest possibly\nC*\n-norm gives its enveloping\nC*\n-algebra, called the group\nC*\n-algebra\nC*(G)\nof G. (Any\nC*\n-norm on\nL1(G)\nis bounded by the\nL1\nnorm, therefore their supremum exists.)\nGiven any abelian\nC*\n-algebra A, the Gelfand transform gives an isomorphism between A and\nC0(A^)\n, where\nA^\nis the multiplicative linear functionals, i.e. one-dimensional representations, on A with the weak-* topology.\nThe map is simply given by\nIt turns out that the multiplicative linear functionals of\nC*(G)\n, after suitable identification, are exactly the characters of G, and the Gelfand transform, when restricted to the dense subset\nL1(G)\nis the Fourier–Pontryagin transform.\n\n## Compact non-abelian groups\nThe Fourier transform can also be defined for functions on a non-abelian group, provided that the group is [compact](https://bliptext.com/articles/compact-space).\nRemoving the assumption that the underlying group is abelian, irreducible unitary representations need not always be one-dimensional.\nThis means the Fourier transform on a non-abelian group takes values as Hilbert space operators.\nThe Fourier transform on compact groups is a major tool in [representation theory](https://bliptext.com/articles/representation-theory) and [non-commutative harmonic analysis](https://bliptext.com/articles/non-commutative-harmonic-analysis).\nLet G be a compact [Hausdorff](https://bliptext.com/articles/hausdorff-space) [topological group](https://bliptext.com/articles/topological-group).\nLet\nΣ\ndenote the collection of all isomorphism classes of finite-dimensional irreducible [unitary representations](https://bliptext.com/articles/unitary-representation), along with a definite choice of representation\nU\non the [Hilbert space](https://bliptext.com/articles/hilbert-space)\nHσ\nof finite dimension\ndσ\nfor each\nσ ∈ Σ\n. If μ is a finite [Borel measure](https://bliptext.com/articles/borel-measure) on G, then the Fourier–Stieltjes transform of μ is the operator on\nHσ\ndefined by\nwhere\nU\nis the complex-conjugate representation of\nU(σ)\nacting on\nHσ\n. If μ is [absolutely continuous](https://bliptext.com/articles/absolutely-continuous) with respect to the [left-invariant probability measure](https://bliptext.com/articles/haar-measure) λ on G, [represented](https://bliptext.com/articles/radon-nikodym-theorem) as\nfor some\nf ∈ L1(λ)\n, one identifies the Fourier transform of f with the Fourier–Stieltjes transform of μ.\nThe mapping\ndefines an isomorphism between the [Banach space](https://bliptext.com/articles/banach-space)\nM(G)\nof finite Borel measures (see [rca space](https://bliptext.com/articles/rca-space)) and a closed subspace of the Banach space\nC∞(Σ)\nconsisting of all sequences\nindexed by\nΣ\nof (bounded) linear operators\nEσ : Hσ → Hσ\nfor which the norm\nis finite.\nThe \"[convolution theorem](https://bliptext.com/articles/convolution-theorem)\" asserts that, furthermore, this isomorphism of Banach spaces is in fact an isometric isomorphism of [C*-algebras](https://bliptext.com/articles/c-algebra) into a subspace of\nC∞(Σ)\n. Multiplication on\nM(G)\nis given by [convolution](https://bliptext.com/articles/convolution) of measures and the involution * defined by\nand\nC∞(Σ)\nhas a natural\nC*\n-algebra structure as Hilbert space operators.\nThe [Peter–Weyl theorem](https://bliptext.com/articles/peter-weyl-theorem) holds, and a version of the Fourier inversion formula ([Plancherel's theorem](https://bliptext.com/articles/plancherel-s-theorem)) follows: if\nf ∈ L2(G)\n, then\nwhere the summation is understood as convergent in the\nL2\nsense.\nThe generalization of the Fourier transform to the noncommutative situation has also in part contributed to the development of [noncommutative geometry](https://bliptext.com/articles/noncommutative-geometry).\nIn this context, a categorical generalization of the Fourier transform to noncommutative groups is [Tannaka–Krein duality](https://bliptext.com/articles/tannaka-krein-duality), which replaces the group of characters with the category of representations.\nHowever, this loses the connection with harmonic functions.\n\n# Alternatives\nIn [signal processing](https://bliptext.com/articles/signal-processing) terms, a function (of time) is a representation of a signal with perfect time resolution, but no frequency information, while the Fourier transform has perfect frequency resolution, but no time information: the magnitude of the Fourier transform at a point is how much frequency content there is, but location is only given by phase (argument of the Fourier transform at a point), and [standing waves](https://bliptext.com/articles/standing-wave) are not localized in time – a sine wave continues out to infinity, without decaying.\nThis limits the usefulness of the Fourier transform for analyzing signals that are localized in time, notably [transients](https://bliptext.com/articles/transient-acoustics), or any signal of finite extent.\nAs alternatives to the Fourier transform, in [time–frequency analysis](https://bliptext.com/articles/time-frequency-analysis), one uses time–frequency transforms or time–frequency distributions to represent signals in a form that has some time information and some frequency information – by the uncertainty principle, there is a trade-off between these.\nThese can be generalizations of the Fourier transform, such as the [short-time Fourier transform](https://bliptext.com/articles/short-time-fourier-transform), [fractional Fourier transform](https://bliptext.com/articles/fractional-fourier-transform), Synchrosqueezing Fourier transform, or other functions to represent signals, as in [wavelet transforms](https://bliptext.com/articles/wavelet-transform) and [chirplet transforms](https://bliptext.com/articles/chirplet-transform), with the wavelet analog of the (continuous) Fourier transform being the [continuous wavelet transform](https://bliptext.com/articles/continuous-wavelet-transform).\n\n# Applications\nLinear operations performed in one domain (time or frequency) have corresponding operations in the other domain, which are sometimes easier to perform.\nThe operation of [differentiation](https://bliptext.com/articles/derivative) in the time domain corresponds to multiplication by the frequency, so some [differential equations](https://bliptext.com/articles/differential-equation) are easier to analyze in the frequency domain.\nAlso, [convolution](https://bliptext.com/articles/convolution) in the time domain corresponds to ordinary multiplication in the frequency domain (see [Convolution theorem](https://bliptext.com/articles/convolution-theorem)).\nAfter performing the desired operations, transformation of the result can be made back to the time domain.\n[Harmonic analysis](https://bliptext.com/articles/harmonic-analysis) is the systematic study of the relationship between the frequency and time domains, including the kinds of functions or operations that are \"simpler\" in one or the other, and has deep connections to many areas of modern mathematics.\n# [Some problems, such as certain differential equations, become easier to solve when the Fourier transform is applied. In that case the solution to the original problem is recovered using the inverse Fourier transform. | upload.wikimedia.org/wikipedia/commons/f/fd/Commutative///diagram///illustrating///problem///solving///via///the///Fourier///transform.svg]\n\n## Analysis of differential equations\nPerhaps the most important use of the Fourier transformation is to solve [partial differential equations](https://bliptext.com/articles/partial-differential-equation).\nMany of the equations of the mathematical physics of the nineteenth century can be treated this way.\nFourier studied the heat equation, which in one dimension and in dimensionless units is\nThe example we will give, a slightly more difficult one, is the wave equation in one dimension,\nAs usual, the problem is not to find a solution: there are infinitely many.\nThe problem is that of the so-called \"boundary problem\": find a solution which satisfies the \"boundary conditions\"\nHere, f and g are given functions.\nFor the heat equation, only one boundary condition can be required (usually the first one).\nBut for the wave equation, there are still infinitely many solutions y which satisfy the first boundary condition.\nBut when one imposes both conditions, there is only one possible solution.\nIt is easier to find the Fourier transform ŷ of the solution than to find the solution directly.\nThis is because the Fourier transformation takes differentiation into multiplication by the Fourier-dual variable, and so a partial differential equation applied to the original function is transformed into multiplication by polynomial functions of the dual variables applied to the transformed function.\nAfter ŷ is determined, we can apply the inverse Fourier transformation to find y.\nFourier's method is as follows.\nFirst, note that any function of the forms\nsatisfies the wave equation.\nThese are called the elementary solutions.\nSecond, note that therefore any integral\nsatisfies the wave equation for arbitrary\na+, a−, b+, b−\n. This integral may be interpreted as a continuous linear combination of solutions for the linear equation.\nNow this resembles the formula for the Fourier synthesis of a function.\nIn fact, this is the real inverse Fourier transform of\na±\nand\nb±\nin the variable x.\nThe third step is to examine how to find the specific unknown coefficient functions\na±\nand\nb±\nthat will lead to y satisfying the boundary conditions.\nWe are interested in the values of these solutions at\nt = 0\n. So we will set\nt = 0\n. Assuming that the conditions needed for Fourier inversion are satisfied, we can then find the Fourier sine and cosine transforms (in the variable x) of both sides and obtain\nand\nSimilarly, taking the derivative of y with respect to t and then applying the Fourier sine and cosine transformations yields\nand\nThese are four linear equations for the four unknowns\na±\nand\nb±\n, in terms of the Fourier sine and cosine transforms of the boundary conditions, which are easily solved by elementary algebra, provided that these transforms can be found.\nIn summary, we chose a set of elementary solutions, parametrized by ξ, of which the general solution would be a (continuous) linear combination in the form of an integral over the parameter ξ.\nBut this integral was in the form of a Fourier integral.\nThe next step was to express the boundary conditions in terms of these integrals, and set them equal to the given functions f and g. But these expressions also took the form of a Fourier integral because of the properties of the Fourier transform of a derivative.\nThe last step was to exploit Fourier inversion by applying the Fourier transformation to both sides, thus obtaining expressions for the coefficient functions\na±\nand\nb±\nin terms of the given boundary conditions f and g.\nFrom a higher point of view, Fourier's procedure can be reformulated more conceptually.\nSince there are two variables, we will use the Fourier transformation in both x and t rather than operate as Fourier did, who only transformed in the spatial variables.\nNote that ŷ must be considered in the sense of a distribution since\ny(x, t)\nis not going to be\nL1\ni2πξ\nand differentiation with respect to t to multiplication by\ni2πf\nwhere f is the frequency.\nThen the wave equation becomes an algebraic equation in ŷ:\nThis is equivalent to requiring\nŷ(ξ, f) = 0\nunless\nξ = ±f\n. Right away, this explains why the choice of elementary solutions we made earlier worked so well: obviously\nf̂ = δ(ξ ± f)\nwill be solutions.\nApplying Fourier inversion to these delta functions, we obtain the elementary solutions we picked earlier.\nBut from the higher point of view, one does not pick elementary solutions, but rather considers the space of all distributions which are supported on the (degenerate) conic\nξ − f = 0\n.\nWe may as well consider the distributions supported on the conic that are given by distributions of one variable on the line\nξ = f\nplus distributions on the line\nas follows: if Φ is any test function,\nwhere\ns+\n, and\ns−\n, are distributions of one variable.\nThen Fourier inversion gives, for the boundary conditions, something very similar to what we had more concretely above (put\nΦ(ξ, f) = ei2π(xξ+tf)\n, which is clearly of polynomial growth):\nand\nNow, as before, applying the one-variable Fourier transformation in the variable x to these functions of x yields two equations in the two unknown distributions\ns±\n(which can be taken to be ordinary functions if the boundary conditions are\nL1\nor\nL2\n).\nFrom a calculational point of view, the drawback of course is that one must first calculate the Fourier transforms of the boundary conditions, then assemble the solution from these, and then calculate an inverse Fourier transform.\nClosed form formulas are rare, except when there is some geometric symmetry that can be exploited, and the numerical calculations are difficult because of the oscillatory nature of the integrals, which makes convergence slow and hard to estimate.\nFor practical calculations, other methods are often used.\nThe twentieth century has seen the extension of these methods to all linear partial differential equations with polynomial coefficients, and by extending the notion of Fourier transformation to include Fourier integral operators, some non-linear equations as well.\n\n## Fourier-transform spectroscopy\nThe Fourier transform is also used in [nuclear magnetic resonance](https://bliptext.com/articles/nuclear-magnetic-resonance) (NMR) and in other kinds of [spectroscopy](https://bliptext.com/articles/spectroscopy), e.g. infrared ([FTIR](https://bliptext.com/articles/fourier-transform-infrared-spectroscopy)).\nIn NMR an exponentially shaped free induction decay (FID) signal is acquired in the time domain and Fourier-transformed to a Lorentzian line-shape in the frequency domain.\nThe Fourier transform is also used in [magnetic resonance imaging](https://bliptext.com/articles/magnetic-resonance-imaging) (MRI) and [mass spectrometry](https://bliptext.com/articles/mass-spectrometry).\n\n## Quantum mechanics\nThe Fourier transform is useful in [quantum mechanics](https://bliptext.com/articles/quantum-mechanics) in at least two different ways.\nTo begin with, the basic conceptual structure of quantum mechanics postulates the existence of pairs of [complementary variables](https://bliptext.com/articles/complementary-variables), connected by the [Heisenberg uncertainty principle](https://bliptext.com/articles/heisenberg-uncertainty-principle).\nFor example, in one dimension, the spatial variable q of, say, a particle, can only be measured by the quantum mechanical \"[position operator](https://bliptext.com/articles/position-operator)\" at the cost of losing information about the momentum p of the particle.\nTherefore, the physical state of the particle can either be described by a function, called \"the wave function\", of q or by a function of p but not by a function of both variables.\nThe variable p is called the conjugate variable to q. In classical mechanics, the physical state of a particle (existing in one dimension, for simplicity of exposition) would be given by assigning definite values to both p and q simultaneously.\nThus, the set of all possible physical states is the two-dimensional real vector space with a p-axis and a q-axis called the [phase space](https://bliptext.com/articles/phase-space).\nIn contrast, quantum mechanics chooses a polarisation of this space in the sense that it picks a subspace of one-half the dimension, for example, the q-axis alone, but instead of considering only points, takes the set of all complex-valued \"wave functions\" on this axis.\nNevertheless, choosing the p-axis is an equally valid polarisation, yielding a different representation of the set of possible physical states of the particle.\nBoth representations of the wavefunction are related by a Fourier transform, such that\nor, equivalently,\nPhysically realisable states are\nL2\n, and so by the [Plancherel theorem](https://bliptext.com/articles/plancherel-theorem), their Fourier transforms are also\nL2\n. (Note that since q is in units of distance and p is in units of momentum, the presence of the Planck constant in the exponent makes the exponent [dimensionless](https://bliptext.com/articles/nondimensionalization), as it should be.)\nTherefore, the Fourier transform can be used to pass from one way of representing the state of the particle, by a wave function of position, to another way of representing the state of the particle: by a wave function of momentum.\nInfinitely many different polarisations are possible, and all are equally valid.\nBeing able to transform states from one representation to another by the Fourier transform is not only convenient but also the underlying reason of the Heisenberg [uncertainty principle](https://bliptext.com/articles/).\nThe other use of the Fourier transform in both quantum mechanics and [quantum field theory](https://bliptext.com/articles/quantum-field-theory) is to solve the applicable wave equation.\nIn non-relativistic quantum mechanics, [Schrödinger's equation](https://bliptext.com/articles/schr-dinger-s-equation) for a time-varying wave function in one-dimension, not subject to external forces, is\nThis is the same as the heat equation except for the presence of the imaginary unit i. Fourier methods can be used to solve this equation.\nIn the presence of a potential, given by the potential energy function\nV(x)\n, the equation becomes\nThe \"elementary solutions\", as we referred to them above, are the so-called \"stationary states\" of the particle, and Fourier's algorithm, as described above, can still be used to solve the boundary value problem of the future evolution of ψ given its values for\n. Neither of these approaches is of much practical use in quantum mechanics.\nBoundary value problems and the time-evolution of the wave function is not of much practical interest: it is the stationary states that are most important.\nIn relativistic quantum mechanics, Schrödinger's equation becomes a wave equation as was usual in classical physics, except that complex-valued waves are considered.\nA simple example, in the absence of interactions with other particles or fields, is the free one-dimensional Klein–Gordon–Schrödinger–Fock equation, this time in dimensionless units,\nThis is, from the mathematical point of view, the same as the wave equation of classical physics solved above (but with a complex-valued wave, which makes no difference in the methods).\nThis is of great use in quantum field theory: each separate Fourier component of a wave can be treated as a separate harmonic oscillator and then quantized, a procedure known as \"second quantization\".\nFourier methods have been adapted to also deal with non-trivial interactions.\nFinally, the [number operator](https://bliptext.com/articles/quantum-harmonic-oscillator) of the [quantum harmonic oscillator](https://bliptext.com/articles/quantum-harmonic-oscillator) can be interpreted, for example via the [Mehler kernel](https://bliptext.com/articles/mehler-kernel), as the [generator](https://bliptext.com/articles/symmetry-in-quantum-mechanics) of the [Fourier transform](https://bliptext.com/articles/) \\mathcal{F}.\n\n## Signal processing\nThe Fourier transform is used for the spectral analysis of time-series.\nThe subject of statistical signal processing does not, however, usually apply the Fourier transformation to the signal itself.\nEven if a real signal is indeed transient, it has been found in practice advisable to model a signal by a function (or, alternatively, a stochastic process) which is stationary in the sense that its characteristic properties are constant over all time.\nThe Fourier transform of such a function does not exist in the usual sense, and it has been found more useful for the analysis of signals to instead take the Fourier transform of its autocorrelation function.\nThe autocorrelation function R of a function f is defined by\nThis function is a function of the time-lag τ elapsing between the values of f to be correlated.\nFor most functions f that occur in practice, R is a bounded even function of the time-lag τ and for typical noisy signals it turns out to be uniformly continuous with a maximum at\n.\nThe autocorrelation function, more properly called the autocovariance function unless it is normalized in some appropriate fashion, measures the strength of the correlation between the values of f separated by a time lag.\nThis is a way of searching for the correlation of f with its own past.\nIt is useful even for other statistical tasks besides the analysis of signals.\nFor example, if\nf(t)\nrepresents the temperature at time t, one expects a strong correlation with the temperature at a time lag of 24 hours.\nIt possesses a Fourier transform,\nThis Fourier transform is called the [power spectral density](https://bliptext.com/articles/spectral-density) function of f. (Unless all periodic components are first filtered out from f, this integral will diverge, but it is easy to filter out such periodicities.)\nThe power spectrum, as indicated by this density function P, measures the amount of variance contributed to the data by the frequency ξ.\nIn electrical signals, the variance is proportional to the average power (energy per unit time), and so the power spectrum describes how much the different frequencies contribute to the average power of the signal.\nThis process is called the spectral analysis of time-series and is analogous to the usual analysis of variance of data that is not a time-series ([ANOVA](https://bliptext.com/articles/anova)).\nKnowledge of which frequencies are \"important\" in this sense is crucial for the proper design of filters and for the proper evaluation of measuring apparatuses.\nIt can also be useful for the scientific analysis of the phenomena responsible for producing the data.\nThe power spectrum of a signal can also be approximately measured directly by measuring the average power that remains in a signal after all the frequencies outside a narrow band have been filtered out.\nSpectral analysis is carried out for visual signals as well.\nThe power spectrum ignores all phase relations, which is good enough for many purposes, but for video signals other types of spectral analysis must also be employed, still using the Fourier transform as a tool.\n\n# Other notations\nOther common notations for \\hat f(\\xi) include:\nIn the sciences and engineering it is also common to make substitutions like these:\nSo the transform pair can become\nA disadvantage of the capital letter notation is when expressing a transform such as or which become the more awkward and\nIn some contexts such as particle physics, the same symbol f may be used for both for a function as well as it Fourier transform, with the two only distinguished by their [argument](https://bliptext.com/articles/argument-of-a-function) I.e. would refer to the Fourier transform because of the momentum [argument](https://bliptext.com/articles/argument-of-a-function), while would refer to the original function because of the positional [argument](https://bliptext.com/articles/argument-of-a-function).\nAlthough tildes may be used as in \\tilde{f} to indicate Fourier transforms, tildes may also be used to indicate a modification of a quantity with a more [Lorentz invariant](https://bliptext.com/articles/lorentz-invariant) form, such as, so care must be taken.\nSimilarly, \\hat f often denotes the [Hilbert transform](https://bliptext.com/articles/hilbert-transform) of f.\nThe interpretation of the complex function\nf̂(ξ)\nmay be aided by expressing it in [polar coordinate](https://bliptext.com/articles/polar-coordinate) form\nin terms of the two real functions\nA(ξ)\nand\nφ(ξ)\nwhere:\nis the [amplitude](https://bliptext.com/articles/amplitude) and\nis the [phase](https://bliptext.com/articles/phase-waves) (see [arg function](https://bliptext.com/articles/arg-mathematics)).\nThen the inverse transform can be written:\nwhich is a recombination of all the frequency components of\nf(x)\n. Each component is a complex [sinusoid](https://bliptext.com/articles/sinusoid) of the form\ne2πixξ\nwhose amplitude is\nA(ξ)\nand whose initial [phase angle](https://bliptext.com/articles/phase-waves) (at\nx = 0\n) is\nφ(ξ)\n.\nThe Fourier transform may be thought of as a mapping on function spaces.\nThis mapping is here denoted and\n(f)\nis used to denote the Fourier transform of the function f. This mapping is linear, which means that can also be seen as a linear transformation on the function space and implies that the standard notation in linear algebra of applying a linear transformation to a vector (here the function\nf\n) can be used to write\nf\ninstead of\n(f)\n. Since the result of applying the Fourier transform is again a function, we can be interested in the value of this function evaluated at the value ξ for its variable, and this is denoted either as\nf(ξ)\nor as\n( f)(ξ)\n. Notice that in the former case, it is implicitly understood that is applied first to f and then the resulting function is evaluated at ξ, not the other way around.\nIn mathematics and various applied sciences, it is often necessary to distinguish between a function f and the value of f when its variable equals x, denoted\nf(x)\n. This means that a notation like\n(f(x))\nformally can be interpreted as the Fourier transform of the values of f at x. Despite this flaw, the previous notation appears frequently, often when a particular function or a function of a particular variable is to be transformed.\nFor example,\nis sometimes used to express that the Fourier transform of a [rectangular function](https://bliptext.com/articles/rectangular-function) is a [sinc function](https://bliptext.com/articles/sinc-function), or\nis used to express the shift property of the Fourier transform.\nNotice, that the last example is only correct under the assumption that the transformed function is a function of x, not of\nx0\n.\nAs discussed above, the [characteristic function](https://bliptext.com/articles/characteristic-function-probability-theory) of a random variable is the same as the [Fourier–Stieltjes transform](https://bliptext.com/articles/) of its distribution measure, but in this context it is typical to take a different convention for the constants.\nTypically characteristic function is defined\nAs in the case of the \"non-unitary angular frequency\" convention above, the factor of 2π appears in neither the normalizing constant nor the exponent.\nUnlike any of the conventions appearing above, this convention takes the opposite sign in the exponent.\n\n# Computation methods\nThe appropriate computation method largely depends how the original mathematical function is represented and the desired form of the output function.\nIn this section we consider both functions of a continuous variable, f(x), and functions of a discrete variable (i.e. ordered pairs of x and f values).\nFor discrete-valued x, the transform integral becomes a summation of sinusoids, which is still a continuous function of frequency (\\xi or \\omega).\nWhen the sinusoids are harmonically-related (i.e. when the x-values are spaced at integer multiples of an interval), the transform is called [discrete-time Fourier transform](https://bliptext.com/articles/discrete-time-fourier-transform) (DTFT).\n\n## Discrete Fourier transforms and fast Fourier transforms\nSampling the DTFT at equally-spaced values of frequency is the most common modern method of computation.\nEfficient procedures, depending on the frequency resolution needed, are described at.\nThe [discrete Fourier transform](https://bliptext.com/articles/discrete-fourier-transform) (DFT), used there, is usually computed by a [fast Fourier transform](https://bliptext.com/articles/fast-fourier-transform) (FFT) algorithm.\n\n## Analytic integration of closed-form functions\nTables of [closed-form](https://bliptext.com/articles/closed-form-expression) Fourier transforms, such as and, are created by mathematically evaluating the Fourier analysis integral (or summation) into another [closed-form](https://bliptext.com/articles/closed-form-expression) function of frequency (\\xi or \\omega).\nWhen mathematically possible, this provides a transform for a continuum of frequency values.\nMany computer algebra systems such as [Matlab](https://bliptext.com/articles/matlab) and [Mathematica](https://bliptext.com/articles/mathematica) that are capable of [symbolic integration](https://bliptext.com/articles/symbolic-integration) are capable of computing Fourier transforms analytically.\nFor example, to compute the Fourier transform of\ncos(6πt) e−πt 2\none might enter the command integrate cos(6*pi*t) exp(−pi*t^2) exp(-i*2*pi*f*t) from -inf to inf into [Wolfram Alpha](https://bliptext.com/articles/wolfram-alpha).\n\n## Numerical integration of closed-form continuous functions\nDiscrete sampling of the Fourier transform can also be done by [numerical integration](https://bliptext.com/articles/numerical-integration) of the definition at each value of frequency for which transform is desired.\nThe numerical integration approach works on a much broader class of functions than the analytic approach.\n\n## Numerical integration of a series of ordered pairs\nIf the input function is a series of ordered pairs, numerical integration reduces to just a summation over the set of data pairs.\nThe DTFT is a common subcase of this more general situation.\n\n# Tables of important Fourier transforms\nThe following tables record some closed-form Fourier transforms.\nFor functions\nf(x)\nand\ng(x)\ndenote their Fourier transforms by\nf̂\nand\nĝ\n. Only the three most common conventions are included.\nIt may be useful to notice that entry 105 gives a relationship between the Fourier transform of a function and the original function, which can be seen as relating the Fourier transform and its inverse.\n\n## Functional relationships, one-dimensional\nThe Fourier transforms in this table may be found in or.\n\n## Square-integrable functions, one-dimensional\nThe Fourier transforms in this table may be found in, , or.\n\n## Distributions, one-dimensional\nThe Fourier transforms in this table may be found in or.\n\n## Two-dimensional functions\n\n## Formulas for general \n\nn\n\n-dimensional functions\n\n# Citations","current_revision":"14532de2-b6cb-4c44-9bb4-3e67c5ab0fe1","created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T17:37:37.735Z","updated_at":"2025-03-08T02:32:07.495Z","search_vector":"'fourier':1 'transform':2","search_content":" \n \n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), the **Fourier transform** (**FT**) is an [integral transform](https://bliptext.com/articles/integral-transform) that takes a [function](https://bliptext.com/articles/function-mathematics) as input and outputs another [function](https://bliptext.com/articles/function-mathematics) that describes the extent to which various frequencies are present in the original [function](https://bliptext.com/articles/function-mathematics).\nThe output of the transform is a [complex](https://bliptext.com/articles/complex-number)-valued function of frequency.\nThe term Fourier transform refers to both this complex-valued function and the [mathematical operation](https://bliptext.com/articles/operation-mathematics).\nWhen a distinction needs to be made, the output of the operation is sometimes called the [frequency domain](https://bliptext.com/articles/frequency-domain) representation of the original function.\nThe Fourier transform is analogous to decomposing the [sound](https://bliptext.com/articles/sound) of a musical [chord](https://bliptext.com/articles/chord-music) into the [intensities](https://bliptext.com/articles/sound-intensity) of its constituent [pitches](https://bliptext.com/articles/pitch-music).\nFunctions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa, a phenomenon known as the [uncertainty principle](https://bliptext.com/articles/).\nThe [critical](https://bliptext.com/articles/critical-point-mathematics) case for this principle is the [Gaussian function](https://bliptext.com/articles/gaussian-function), of substantial importance in [probability theory](https://bliptext.com/articles/probability-theory) and [statistics](https://bliptext.com/articles/statistics) as well as in the study of physical phenomena exhibiting [normal distribution](https://bliptext.com/articles/normal-distribution) (e.g., [diffusion](https://bliptext.com/articles/diffusion)).\nThe Fourier transform of a Gaussian function is another Gaussian function.\n[Joseph Fourier](https://bliptext.com/articles/joseph-fourier) introduced [sine and cosine transforms](https://bliptext.com/articles/sine-and-cosine-transforms) (which [correspond to the imaginary and real components](https://bliptext.com/articles/sine-and-cosine-transforms) of the modern Fourier transform) in his study of [heat transfer](https://bliptext.com/articles/heat-transfer), where Gaussian functions appear as solutions of the [heat equation](https://bliptext.com/articles/heat-equation).\nThe Fourier transform can be formally defined as an [improper](https://bliptext.com/articles/improper-integral) [Riemann integral](https://bliptext.com/articles/riemann-integral), making it an integral transform, although this definition is not suitable for many applications requiring a more sophisticated integration theory.\nFor example, many relatively simple applications use the [Dirac delta function](https://bliptext.com/articles/dirac-delta-function), which can be treated formally as if it were a function, but the justification requires a mathematically more sophisticated viewpoint.\nThe Fourier transform can also be generalized to functions of several variables on [Euclidean space](https://bliptext.com/articles/euclidean-space), sending a function of 3-dimensional 'position space' to a function of 3-dimensional momentum (or a function of space and time to a function of [4-momentum](https://bliptext.com/articles/4-momentum)).\nThis idea makes the spatial Fourier transform very natural in the study of waves, as well as in [quantum mechanics](https://bliptext.com/articles/quantum-mechanics), where it is important to be able to represent wave solutions as functions of either position or momentum and sometimes both.\nIn general, functions to which Fourier methods are applicable are complex-valued, and possibly [vector-valued](https://bliptext.com/articles/vector-valued-function).\nStill further generalization is possible to functions on [groups](https://bliptext.com/articles/group-mathematics), which, besides the original Fourier transform on [[Real number#Arithmetic|\nR\n]] or\nRn\n, notably includes the [discrete-time Fourier transform](https://bliptext.com/articles/discrete-time-fourier-transform) (DTFT, group =\nZ\n), the [discrete Fourier transform](https://bliptext.com/articles/discrete-fourier-transform) (DFT, group = [[cyclic group|\nZ mod N\n]]) and the [Fourier series](https://bliptext.com/articles/fourier-series) or circular Fourier transform (group =\nS1\n, the unit circle ≈ closed finite interval with endpoints identified). The latter is routinely employed to handle [periodic functions](https://bliptext.com/articles/periodic-function). The [fast Fourier transform](https://bliptext.com/articles/fast-fourier-transform) (FFT) is an algorithm for computing the DFT.\n# [An example application of the Fourier transform is determining the constituent pitches in a musical waveform. This image is the result of applying a constant-Q transform (a Fourier-related transform) to the waveform of a C major piano chord. The first three peaks on the left correspond to the frequencies of the fundamental frequency of the chord (C, E, G). The remaining smaller peaks are higher-frequency overtones of the fundamental pitches. A pitch detection algorithm could use the relative intensity of these peaks to infer which notes the pianist pressed. | upload.wikimedia.org/wikipedia/commons/3/31/CQT-piano-chord.png]\n\n# Definition\nThe Fourier transform is an analysis process, decomposing a complex-valued function into its constituent frequencies and their amplitudes.\nThe inverse process is synthesis, which recreates from its transform.\nWe can start with an analogy, the **[Fourier series](https://bliptext.com/articles/fourier-series)**, which analyzes on a bounded interval for some positive real number P. The constituent frequencies are a discrete set of harmonics at frequencies whose amplitude and phase are given by the **analysis formula****:******The actual **[Fourier series](https://bliptext.com/articles/fourier-series)** is the **synthesis formula****:******On an unbounded interval, P\\to\\infty, the constituent frequencies are a continuum****:**** and c_n is replaced by a function****:****\nEvaluating for all values of \\xi produces the frequency-domain function.\nThe integral can diverge at some frequencies.\n(see ) But it converges for ** all ** frequencies when f(x) decays with** all **derivatives as **:**.\n(See [Schwartz function](https://bliptext.com/articles/schwartz-function)).\nBy the [Riemann–Lebesgue lemma](https://bliptext.com/articles/riemann-lebesgue-lemma), the transformed function \\widehat f also decays with all derivatives.\nThe complex number, in polar coordinates, conveys both [amplitude](https://bliptext.com/articles/amplitude) and [phase](https://bliptext.com/articles/phase-offset) of frequency \\xi.\nThe intuitive interpretation of is that the effect of multiplying f(x) by is to subtract \\xi from every frequency component of function f(x).\nOnly the component that was at frequency \\xi can produce a non-zero value of the infinite integral, because (at least formally) all the other shifted components are oscillatory and integrate to zero.\n(see )\nThe corresponding synthesis formula is:\nis a representation of f(x) as a weighted summation of complex exponential functions.\nThis is also known as the [Fourier inversion theorem](https://bliptext.com/articles/fourier-inversion-theorem), and was first introduced in [Fourier's](https://bliptext.com/articles/joseph-fourier) Analytical Theory of Heat.\nThe functions f and \\widehat{f} are referred to as a **Fourier transform pair**.\nA common notation for designating transform pairs is**:**\nfor example\n\n## Definition for Lebesgue integrable functions\nUntil now, we have been dealing with Schwartz functions, which decay rapidly at infinity, with all derivatives.\nThis excludes many functions of practical importance from the definition, such as the [rect function](https://bliptext.com/articles/rect-function).\nA [measurable function](https://bliptext.com/articles/measurable-function) is called (Lebesgue) integrable if the [Lebesgue integral](https://bliptext.com/articles/lebesgue-integral) of its absolute value is finite:\nTwo measurable functions are equivalent if they are equal except on a set of measure zero.\nThe set of all equivalence classes of integrable functions is denoted.\nThen:\nThe integral is well-defined for all because of the assumption.\n(It can be shown that the function is bounded and [uniformly continuous](https://bliptext.com/articles/uniformly-continuous) in the frequency domain, and moreover, by the [Riemann–Lebesgue lemma](https://bliptext.com/articles/riemann-lebesgue-lemma), it is zero at infinity.)\nHowever, the class of Lebesgue integrable functions is not ideal from the point of view of the Fourier transform because there is no easy characterization of the image, and thus no easy characterization of the inverse transform.\n\n## Unitarity and definition for square integrable functions\nWhile defines the Fourier transform for (complex-valued) functions in, it is easy to see that it is not well-defined for other integrability classes, most importantly.\nFor functions in, and with the conventions of , the Fourier transform is a [unitary operator](https://bliptext.com/articles/unitary-operator) with respect to the Hilbert inner product on , restricted to the dense subspace of integrable functions.\nTherefore, it admits a unique continuous extension to a unitary operator on, also called the Fourier transform.\nThis extension is important in part because the Fourier transform preserves the space so that, unlike the case of L^1, the Fourier transform and inverse transform are on the same footing, being transformations of the same space of functions to itself.\nImportantly, for functions in L^2, the Fourier transform is no longer given by (interpreted as a Lebesgue integral).\nFor example, the function is in L^2 but not L^1, so the integral diverges.\nIn such cases, the Fourier transform can be obtained explicitly by regularizing the integral, and then passing to a limit.\nIn practice, the integral is often regarded as an [improper integral](https://bliptext.com/articles/improper-integral) instead of a proper Lebesgue integral, but sometimes for convergence one needs to use [weak limit](https://bliptext.com/articles/weak-limit) or [principal value](https://bliptext.com/articles/cauchy-principal-value) instead of the (pointwise) limits implicit in an [improper integral](https://bliptext.com/articles/improper-integral).\nand each gives three rigorous ways of extending the Fourier transform to square integrable functions using this procedure.\nThe conventions chosen in this article are those of [harmonic analysis](https://bliptext.com/articles/harmonic-analysis), and are characterized as the unique conventions such that the Fourier transform is both [unitary](https://bliptext.com/articles/unitary-operator) on\nL2\nand an algebra homomorphism from\nL1\nto\nL∞\n, without renormalizing the Lebesgue measure.\n\n## Angular frequency (ω)\nWhen the independent variable (x) represents time (often denoted by t), the transform variable (\\xi) represents [frequency](https://bliptext.com/articles/frequency) (often denoted by f).\nFor example, if time is measured in [seconds](https://bliptext.com/articles/second), then frequency is in [hertz](https://bliptext.com/articles/hertz).\nThe Fourier transform can also be written in terms of [angular frequency](https://bliptext.com/articles/angular-frequency), whose units are [radians](https://bliptext.com/articles/radian) per second.\nThe substitution into produces this convention, where function \\widehat f is relabeled\nUnlike the definition, the Fourier transform is no longer a [unitary transformation](https://bliptext.com/articles/unitary-transformation), and there is less symmetry between the formulas for the transform and its inverse.\nThose properties are restored by splitting the 2 \\pi factor evenly between the transform and its inverse, which leads to another convention:\nVariations of all three conventions can be created by conjugating the complex-exponential [kernel](https://bliptext.com/articles/integral-kernel) of both the forward and the reverse transform.\nThe signs must be opposites.\n\n## Extension of the definition\nFor 1 0\nsuch that for any [integer](https://bliptext.com/articles/integer)\nn ≥ 0\n,\nfor some constant C. (In this case, f is supported on\n[−a, a]\n.) This can be expressed by saying that\nf̂\nis an [entire function](https://bliptext.com/articles/entire-function) which is [rapidly decreasing](https://bliptext.com/articles/rapidly-decreasing) in σ (for fixed τ) and of exponential growth in τ (uniformly in σ).\n(If f is not smooth, but only\nL2\n, the statement still holds provided\n. ) The space of such functions of a [complex variable](https://bliptext.com/articles/complex-analysis) is called the Paley—Wiener space. This theorem has been generalised to semisimple [Lie groups](https://bliptext.com/articles/lie-group).\nIf f is supported on the half-line\nt ≥ 0\n, then f is said to be \"causal\" because the [impulse response function](https://bliptext.com/articles/impulse-response-function) of a physically realisable [filter](https://bliptext.com/articles/filter-mathematics) must have this property, as no effect can precede its cause.\n[Paley](https://bliptext.com/articles/raymond-paley) and Wiener showed that then\nf̂\nextends to a [holomorphic function](https://bliptext.com/articles/holomorphic-function) on the complex lower half-plane\nτ 0\n, then\nfor any\nτ a\n, where\nF(s)\nis the Laplace transform of\nf(t)\n.\nThe hypotheses can be weakened, as in the results of Carleson and Hunt, to\nf(t) e−at\nbeing\nL1\n, provided that f be of bounded variation in a closed neighborhood of t (cf. [Dini test](https://bliptext.com/articles/dini-test)), the value of f at t be taken to be the [arithmetic mean](https://bliptext.com/articles/arithmetic-mean) of the left and right limits, and that the integrals be taken in the sense of Cauchy principal values.\nL2\nversions of these inversion formulas are also available.\n\n# Fourier transform on Euclidean space\nThe Fourier transform can be defined in any arbitrary number of dimensions n. As with the one-dimensional case, there are many conventions.\nFor an integrable function\nf(x)\n, this article takes the definition:\nwhere\nx\nand\nξ\nare n-dimensional [vectors](https://bliptext.com/articles/vector-mathematics), and\nx · ξ\nis the [dot product](https://bliptext.com/articles/dot-product) of the vectors.\nAlternatively,\nξ\ncan be viewed as belonging to the [dual vector space](https://bliptext.com/articles/dual-space) \\R^{n\\star}, in which case the dot product becomes the [contraction](https://bliptext.com/articles/tensor-contraction) of\nx\nand\nξ\n, usually written as\n⟨x, ξ⟩\n.\nAll of the basic properties listed above hold for the n-dimensional Fourier transform, as do Plancherel's and Parseval's theorem.\nWhen the function is integrable, the Fourier transform is still uniformly continuous and the [Riemann–Lebesgue lemma](https://bliptext.com/articles/riemann-lebesgue-lemma) holds.\n\n## Uncertainty principle\nGenerally speaking, the more concentrated\nf(x)\nis, the more spread out its Fourier transform\nf̂(ξ)\nmust be.\nIn particular, the scaling property of the Fourier transform may be seen as saying: if we squeeze a function in x, its Fourier transform stretches out in ξ.\nIt is not possible to arbitrarily concentrate both a function and its Fourier transform.\nThe trade-off between the compaction of a function and its Fourier transform can be formalized in the form of an [uncertainty principle](https://bliptext.com/articles/uncertainty-principle) by viewing a function and its Fourier transform as [conjugate variables](https://bliptext.com/articles/conjugate-variables) with respect to the [[symplectic form](https://bliptext.com/articles/symplectic-vector-space)](https://bliptext.com/articles/symplectic-form) on the [time–frequency domain](https://bliptext.com/articles/time-frequency-representation): from the point of view of the [linear canonical transformation](https://bliptext.com/articles/linear-canonical-transformation), the Fourier transform is rotation by 90° in the [time–frequency domain](https://bliptext.com/articles/time-frequency-representation), and preserves the [[symplectic form](https://bliptext.com/articles/symplectic-vector-space)](https://bliptext.com/articles/symplectic-form).\nSuppose\nf(x)\nis an integrable and [square-integrable](https://bliptext.com/articles/square-integrable) function.\nWithout loss of generality, assume that\nf(x)\nis normalized:\nIt follows from the [Plancherel theorem](https://bliptext.com/articles/plancherel-theorem) that\nf̂(ξ)\nis also normalized.\nThe spread around\nmay be measured by the dispersion about zero defined by\nIn probability terms, this is the [second moment](https://bliptext.com/articles/moment-mathematics) of\nabout zero.\nThe uncertainty principle states that, if\nf(x)\nis absolutely continuous and the functions\nx·f(x)\nand\nf(x)\nare square integrable, then\nThe equality is attained only in the case\nwhere\nσ > 0\nis arbitrary and\nC1 = 4√2⁄√σ\nso that f is\nL2\n-normalized.\nIn other words, where f is a (normalized) [Gaussian function](https://bliptext.com/articles/gaussian-function) with variance\nσ2/2π\n, centered at zero, and its Fourier transform is a Gaussian function with variance\nσ−2/2π\n.\nIn fact, this inequality implies that:\nfor any\nx0\n,\nξ0 ∈ R\n.\nIn [quantum mechanics](https://bliptext.com/articles/quantum-mechanics), the [momentum](https://bliptext.com/articles/momentum) and position [wave functions](https://bliptext.com/articles/wave-function) are Fourier transform pairs, up to a factor of the [Planck constant](https://bliptext.com/articles/planck-constant).\nWith this constant properly taken into account, the inequality above becomes the statement of the [Heisenberg uncertainty principle](https://bliptext.com/articles/heisenberg-uncertainty-principle).\nA stronger uncertainty principle is the [Hirschman uncertainty principle](https://bliptext.com/articles/hirschman-uncertainty), which is expressed as:\nwhere\nH(p)\nis the [differential entropy](https://bliptext.com/articles/differential-entropy) of the [probability density function](https://bliptext.com/articles/probability-density-function)\np(x)\nwhere the logarithms may be in any base that is consistent.\nThe equality is attained for a Gaussian, as in the previous case.\n\n## Sine and cosine transforms\nFourier's original formulation of the transform did not use complex numbers, but rather sines and cosines.\nStatisticians and others still use this form.\nAn absolutely integrable function f for which Fourier inversion holds can be expanded in terms of genuine frequencies (avoiding negative frequencies, which are sometimes considered hard to interpret physically ) λ by\nThis is called an expansion as a trigonometric integral, or a Fourier integral expansion.\nThe coefficient functions a and b can be found by using variants of the Fourier cosine transform and the Fourier sine transform (the normalisations are, again, not standardised):\nand\nOlder literature refers to the two transform functions, the Fourier cosine transform, a, and the Fourier sine transform, b.\nThe function f can be recovered from the sine and cosine transform using\ntogether with trigonometric identities.\nThis is referred to as Fourier's integral formula.\n\n## Spherical harmonics\nLet the set of [homogeneous](https://bliptext.com/articles/homogeneous-polynomial) [harmonic](https://bliptext.com/articles/harmonic-function) [polynomials](https://bliptext.com/articles/polynomial) of degree k on\nRn\nbe denoted by\nAk\n. The set\nAk\nconsists of the [solid spherical harmonics](https://bliptext.com/articles/solid-spherical-harmonics) of degree k. The [solid spherical harmonics](https://bliptext.com/articles/solid-spherical-harmonics) play a similar role in higher dimensions to the Hermite polynomials in dimension one.\nSpecifically, if\nf(x) = e−π P(x)\nfor some\nP(x)\nin\nAk\n, then\nf̂(ξ) = i f(ξ)\n. Let the set\nHk\nbe the closure in\nL2(Rn)\nof linear combinations of functions of the form\nf(\nwhere\nP(x)\nis in\nAk\n. The space\nL2(Rn)\nis then a direct sum of the spaces\nHk\nand the Fourier transform maps each space\nHk\nto itself and is possible to characterize the action of the Fourier transform on each space\nHk\n.\nLet\n)P(x)\n(with\nP(x)\nin\nAk\n), then\nwhere\nHere\nJ(n + 2k − 2)/2\ndenotes the [Bessel function](https://bliptext.com/articles/bessel-function) of the first kind with order\nn + 2k − 2⁄2\n. When\nthis gives a useful formula for the Fourier transform of a radial function.\nThis is essentially the [Hankel transform](https://bliptext.com/articles/hankel-transform).\nMoreover, there is a simple recursion relating the cases\nn + 2\nand n allowing to compute, e.g., the three-dimensional Fourier transform of a radial function from the one-dimensional one.\n\n## Restriction problems\nIn higher dimensions it becomes interesting to study restriction problems for the Fourier transform.\nThe Fourier transform of an integrable function is continuous and the restriction of this function to any set is defined.\nBut for a square-integrable function the Fourier transform could be a general class of square integrable functions.\nAs such, the restriction of the Fourier transform of an\nL2(Rn)\nfunction cannot be defined on sets of measure 0.\nIt is still an active area of study to understand restriction problems in\nL\nfor\n1 1\n. In the case that ER is taken to be a cube with side length R, then convergence still holds.\nAnother natural candidate is the Euclidean ball {{math|E{{sub|R}} = {ξ : {{abs|ξ}} 2\nso that the Fourier transform is not defined as a function.\n\n## Tempered distributions\nOne might consider enlarging the domain of the Fourier transform from\nL1 + L2\nby considering [generalized functions](https://bliptext.com/articles/generalized-function), or distributions.\nA distribution on\nRn\nis a continuous linear functional on the space\nCc(Rn)\nof compactly supported smooth functions, equipped with a suitable topology.\nThe strategy is then to consider the action of the Fourier transform on\nCc(Rn)\nand pass to distributions by duality.\nThe obstruction to doing this is that the Fourier transform does not map\nCc(Rn)\nto\nCc(Rn)\n. In fact the Fourier transform of an element in\nCc(Rn)\ncan not vanish on an open set; see the above discussion on the uncertainty principle.\nThe right space here is the slightly larger space of [Schwartz functions](https://bliptext.com/articles/schwartz-space).\nThe Fourier transform is an automorphism on the Schwartz space, as a topological vector space, and thus induces an automorphism on its dual, the space of tempered distributions.\nThe tempered distributions include all the integrable functions mentioned above, as well as well-behaved functions of polynomial growth and distributions of compact support.\nFor the definition of the Fourier transform of a tempered distribution, let f and g be integrable functions, and let\nf̂\nand\nĝ\nbe their Fourier transforms respectively.\nThen the Fourier transform obeys the following multiplication formula,\nEvery integrable function f defines (induces) a distribution Tf by the relation\nfor all Schwartz functions φ.\nSo it makes sense to define Fourier transform\nT̂f\nof\nTf\nby\nfor all Schwartz functions φ.\nExtending this to all tempered distributions T gives the general definition of the Fourier transform.\nDistributions can be differentiated and the above-mentioned compatibility of the Fourier transform with differentiation and convolution remains true for tempered distributions.\n\n# Generalizations\n\n## Fourier–Stieltjes transform\nThe Fourier transform of a [finite](https://bliptext.com/articles/finite-measure) [Borel measure](https://bliptext.com/articles/borel-measure) μ on\nRn\nis given by:\nThis transform continues to enjoy many of the properties of the Fourier transform of integrable functions.\nOne notable difference is that the [Riemann–Lebesgue lemma](https://bliptext.com/articles/riemann-lebesgue-lemma) fails for measures.\nIn the case that\n, then the formula above reduces to the usual definition for the Fourier transform of f. In the case that μ is the probability distribution associated to a random variable X, the Fourier–Stieltjes transform is closely related to the [characteristic function](https://bliptext.com/articles/characteristic-function-probability-theory), but the typical conventions in probability theory take\neiξx\ninstead of\ne−i2πξx\n. In the case when the distribution has a [probability density function](https://bliptext.com/articles/probability-density-function) this definition reduces to the Fourier transform applied to the [probability density function](https://bliptext.com/articles/probability-density-function), again with a different choice of constants.\nThe Fourier transform may be used to give a characterization of measures.\n[Bochner's theorem](https://bliptext.com/articles/bochner-s-theorem) characterizes which functions may arise as the Fourier–Stieltjes transform of a positive measure on the circle.\nFurthermore, the [Dirac delta function](https://bliptext.com/articles/dirac-delta-function), although not a function, is a finite Borel measure.\nIts Fourier transform is a constant function (whose specific value depends upon the form of the Fourier transform used).\n\n## Locally compact abelian groups\nThe Fourier transform may be generalized to any locally compact abelian group.\nA [locally compact](https://bliptext.com/articles/locally-compact) [abelian group](https://bliptext.com/articles/abelian-group) is an [abelian group](https://bliptext.com/articles/abelian-group) that is at the same time a [locally compact](https://bliptext.com/articles/locally-compact) [Hausdorff topological space](https://bliptext.com/articles/hausdorff-space) so that the group operation is continuous.\nIf G is a locally compact abelian group, it has a translation invariant measure μ, called [Haar measure](https://bliptext.com/articles/haar-measure).\nFor a locally compact abelian group G, the set of irreducible, i.e. one-dimensional, unitary representations are called its [characters](https://bliptext.com/articles/character-group).\nWith its natural group structure and the topology of uniform convergence on compact sets (that is, the topology induced by the [compact-open topology](https://bliptext.com/articles/compact-open-topology) on the space of all continuous functions from G to the [circle group](https://bliptext.com/articles/circle-group)), the set of characters Ĝ is itself a locally compact abelian group, called the Pontryagin dual of G. For a function f in\nL1(G)\n, its Fourier transform is defined by\nThe Riemann–Lebesgue lemma holds in this case;\nf̂(ξ)\nis a function vanishing at infinity on Ĝ.\nThe Fourier transform on is an example; here T is a locally compact abelian group, and the Haar measure μ on T can be thought of as the Lebesgue measure on [0,1). Consider the representation of T on the complex plane C that is a 1-dimensional complex vector space. There are a group of representations (which are irreducible since C is 1-dim) where for x\\in T.\nThe character of such representation, that is the trace of e_{k}(x) for each x\\in T and k\\in Z, is itself.\nIn the case of representation of finite group, the character table of the group G are rows of vectors such that each row is the character of one irreducible representation of G, and these vectors form an orthonormal basis of the space of class functions that map from G to C by Schur's lemma.\nNow the group T is no longer finite but still compact, and it preserves the orthonormality of character table.\nEach row of the table is the function e_{k}(x) of x\\in T, and the inner product between two class functions (all functions being class functions since T is abelian) is defined as with the normalizing factor |T|=1.\nThe sequence is an orthonormal basis of the space of class functions.\nFor any representation V of a finite group G, \\chi_{v} can be expressed as the span (V_{i} are the irreps of G), such that.\nSimilarly for G = T and,.\nThe Pontriagin dual \\hat{T} is and for, is its Fourier transform for.\n\n## Gelfand transform\nThe Fourier transform is also a special case of [Gelfand transform](https://bliptext.com/articles/gelfand-transform).\nIn this particular context, it is closely related to the Pontryagin duality map defined above.\nGiven an abelian [locally compact](https://bliptext.com/articles/locally-compact-space) [Hausdorff](https://bliptext.com/articles/hausdorff-space) [topological group](https://bliptext.com/articles/topological-group) G, as before we consider space\nL1(G)\n, defined using a Haar measure.\nWith convolution as multiplication,\nL1(G)\nis an abelian [Banach algebra](https://bliptext.com/articles/banach-algebra).\nIt also has an [involution](https://bliptext.com/articles/involution-mathematics) * given by\nTaking the completion with respect to the largest possibly\nC*\n-norm gives its enveloping\nC*\n-algebra, called the group\nC*\n-algebra\nC*(G)\nof G. (Any\nC*\n-norm on\nL1(G)\nis bounded by the\nL1\nnorm, therefore their supremum exists.)\nGiven any abelian\nC*\n-algebra A, the Gelfand transform gives an isomorphism between A and\nC0(A^)\n, where\nA^\nis the multiplicative linear functionals, i.e. one-dimensional representations, on A with the weak-* topology.\nThe map is simply given by\nIt turns out that the multiplicative linear functionals of\nC*(G)\n, after suitable identification, are exactly the characters of G, and the Gelfand transform, when restricted to the dense subset\nL1(G)\nis the Fourier–Pontryagin transform.\n\n## Compact non-abelian groups\nThe Fourier transform can also be defined for functions on a non-abelian group, provided that the group is [compact](https://bliptext.com/articles/compact-space).\nRemoving the assumption that the underlying group is abelian, irreducible unitary representations need not always be one-dimensional.\nThis means the Fourier transform on a non-abelian group takes values as Hilbert space operators.\nThe Fourier transform on compact groups is a major tool in [representation theory](https://bliptext.com/articles/representation-theory) and [non-commutative harmonic analysis](https://bliptext.com/articles/non-commutative-harmonic-analysis).\nLet G be a compact [Hausdorff](https://bliptext.com/articles/hausdorff-space) [topological group](https://bliptext.com/articles/topological-group).\nLet\nΣ\ndenote the collection of all isomorphism classes of finite-dimensional irreducible [unitary representations](https://bliptext.com/articles/unitary-representation), along with a definite choice of representation\nU\non the [Hilbert space](https://bliptext.com/articles/hilbert-space)\nHσ\nof finite dimension\ndσ\nfor each\nσ ∈ Σ\n. If μ is a finite [Borel measure](https://bliptext.com/articles/borel-measure) on G, then the Fourier–Stieltjes transform of μ is the operator on\nHσ\ndefined by\nwhere\nU\nis the complex-conjugate representation of\nU(σ)\nacting on\nHσ\n. If μ is [absolutely continuous](https://bliptext.com/articles/absolutely-continuous) with respect to the [left-invariant probability measure](https://bliptext.com/articles/haar-measure) λ on G, [represented](https://bliptext.com/articles/radon-nikodym-theorem) as\nfor some\nf ∈ L1(λ)\n, one identifies the Fourier transform of f with the Fourier–Stieltjes transform of μ.\nThe mapping\ndefines an isomorphism between the [Banach space](https://bliptext.com/articles/banach-space)\nM(G)\nof finite Borel measures (see [rca space](https://bliptext.com/articles/rca-space)) and a closed subspace of the Banach space\nC∞(Σ)\nconsisting of all sequences\nindexed by\nΣ\nof (bounded) linear operators\nEσ : Hσ → Hσ\nfor which the norm\nis finite.\nThe \"[convolution theorem](https://bliptext.com/articles/convolution-theorem)\" asserts that, furthermore, this isomorphism of Banach spaces is in fact an isometric isomorphism of [C*-algebras](https://bliptext.com/articles/c-algebra) into a subspace of\nC∞(Σ)\n. Multiplication on\nM(G)\nis given by [convolution](https://bliptext.com/articles/convolution) of measures and the involution * defined by\nand\nC∞(Σ)\nhas a natural\nC*\n-algebra structure as Hilbert space operators.\nThe [Peter–Weyl theorem](https://bliptext.com/articles/peter-weyl-theorem) holds, and a version of the Fourier inversion formula ([Plancherel's theorem](https://bliptext.com/articles/plancherel-s-theorem)) follows: if\nf ∈ L2(G)\n, then\nwhere the summation is understood as convergent in the\nL2\nsense.\nThe generalization of the Fourier transform to the noncommutative situation has also in part contributed to the development of [noncommutative geometry](https://bliptext.com/articles/noncommutative-geometry).\nIn this context, a categorical generalization of the Fourier transform to noncommutative groups is [Tannaka–Krein duality](https://bliptext.com/articles/tannaka-krein-duality), which replaces the group of characters with the category of representations.\nHowever, this loses the connection with harmonic functions.\n\n# Alternatives\nIn [signal processing](https://bliptext.com/articles/signal-processing) terms, a function (of time) is a representation of a signal with perfect time resolution, but no frequency information, while the Fourier transform has perfect frequency resolution, but no time information: the magnitude of the Fourier transform at a point is how much frequency content there is, but location is only given by phase (argument of the Fourier transform at a point), and [standing waves](https://bliptext.com/articles/standing-wave) are not localized in time – a sine wave continues out to infinity, without decaying.\nThis limits the usefulness of the Fourier transform for analyzing signals that are localized in time, notably [transients](https://bliptext.com/articles/transient-acoustics), or any signal of finite extent.\nAs alternatives to the Fourier transform, in [time–frequency analysis](https://bliptext.com/articles/time-frequency-analysis), one uses time–frequency transforms or time–frequency distributions to represent signals in a form that has some time information and some frequency information – by the uncertainty principle, there is a trade-off between these.\nThese can be generalizations of the Fourier transform, such as the [short-time Fourier transform](https://bliptext.com/articles/short-time-fourier-transform), [fractional Fourier transform](https://bliptext.com/articles/fractional-fourier-transform), Synchrosqueezing Fourier transform, or other functions to represent signals, as in [wavelet transforms](https://bliptext.com/articles/wavelet-transform) and [chirplet transforms](https://bliptext.com/articles/chirplet-transform), with the wavelet analog of the (continuous) Fourier transform being the [continuous wavelet transform](https://bliptext.com/articles/continuous-wavelet-transform).\n\n# Applications\nLinear operations performed in one domain (time or frequency) have corresponding operations in the other domain, which are sometimes easier to perform.\nThe operation of [differentiation](https://bliptext.com/articles/derivative) in the time domain corresponds to multiplication by the frequency, so some [differential equations](https://bliptext.com/articles/differential-equation) are easier to analyze in the frequency domain.\nAlso, [convolution](https://bliptext.com/articles/convolution) in the time domain corresponds to ordinary multiplication in the frequency domain (see [Convolution theorem](https://bliptext.com/articles/convolution-theorem)).\nAfter performing the desired operations, transformation of the result can be made back to the time domain.\n[Harmonic analysis](https://bliptext.com/articles/harmonic-analysis) is the systematic study of the relationship between the frequency and time domains, including the kinds of functions or operations that are \"simpler\" in one or the other, and has deep connections to many areas of modern mathematics.\n# [Some problems, such as certain differential equations, become easier to solve when the Fourier transform is applied. In that case the solution to the original problem is recovered using the inverse Fourier transform. | upload.wikimedia.org/wikipedia/commons/f/fd/Commutative///diagram///illustrating///problem///solving///via///the///Fourier///transform.svg]\n\n## Analysis of differential equations\nPerhaps the most important use of the Fourier transformation is to solve [partial differential equations](https://bliptext.com/articles/partial-differential-equation).\nMany of the equations of the mathematical physics of the nineteenth century can be treated this way.\nFourier studied the heat equation, which in one dimension and in dimensionless units is\nThe example we will give, a slightly more difficult one, is the wave equation in one dimension,\nAs usual, the problem is not to find a solution: there are infinitely many.\nThe problem is that of the so-called \"boundary problem\": find a solution which satisfies the \"boundary conditions\"\nHere, f and g are given functions.\nFor the heat equation, only one boundary condition can be required (usually the first one).\nBut for the wave equation, there are still infinitely many solutions y which satisfy the first boundary condition.\nBut when one imposes both conditions, there is only one possible solution.\nIt is easier to find the Fourier transform ŷ of the solution than to find the solution directly.\nThis is because the Fourier transformation takes differentiation into multiplication by the Fourier-dual variable, and so a partial differential equation applied to the original function is transformed into multiplication by polynomial functions of the dual variables applied to the transformed function.\nAfter ŷ is determined, we can apply the inverse Fourier transformation to find y.\nFourier's method is as follows.\nFirst, note that any function of the forms\nsatisfies the wave equation.\nThese are called the elementary solutions.\nSecond, note that therefore any integral\nsatisfies the wave equation for arbitrary\na+, a−, b+, b−\n. This integral may be interpreted as a continuous linear combination of solutions for the linear equation.\nNow this resembles the formula for the Fourier synthesis of a function.\nIn fact, this is the real inverse Fourier transform of\na±\nand\nb±\nin the variable x.\nThe third step is to examine how to find the specific unknown coefficient functions\na±\nand\nb±\nthat will lead to y satisfying the boundary conditions.\nWe are interested in the values of these solutions at\nt = 0\n. So we will set\nt = 0\n. Assuming that the conditions needed for Fourier inversion are satisfied, we can then find the Fourier sine and cosine transforms (in the variable x) of both sides and obtain\nand\nSimilarly, taking the derivative of y with respect to t and then applying the Fourier sine and cosine transformations yields\nand\nThese are four linear equations for the four unknowns\na±\nand\nb±\n, in terms of the Fourier sine and cosine transforms of the boundary conditions, which are easily solved by elementary algebra, provided that these transforms can be found.\nIn summary, we chose a set of elementary solutions, parametrized by ξ, of which the general solution would be a (continuous) linear combination in the form of an integral over the parameter ξ.\nBut this integral was in the form of a Fourier integral.\nThe next step was to express the boundary conditions in terms of these integrals, and set them equal to the given functions f and g. But these expressions also took the form of a Fourier integral because of the properties of the Fourier transform of a derivative.\nThe last step was to exploit Fourier inversion by applying the Fourier transformation to both sides, thus obtaining expressions for the coefficient functions\na±\nand\nb±\nin terms of the given boundary conditions f and g.\nFrom a higher point of view, Fourier's procedure can be reformulated more conceptually.\nSince there are two variables, we will use the Fourier transformation in both x and t rather than operate as Fourier did, who only transformed in the spatial variables.\nNote that ŷ must be considered in the sense of a distribution since\ny(x, t)\nis not going to be\nL1\ni2πξ\nand differentiation with respect to t to multiplication by\ni2πf\nwhere f is the frequency.\nThen the wave equation becomes an algebraic equation in ŷ:\nThis is equivalent to requiring\nŷ(ξ, f) = 0\nunless\nξ = ±f\n. Right away, this explains why the choice of elementary solutions we made earlier worked so well: obviously\nf̂ = δ(ξ ± f)\nwill be solutions.\nApplying Fourier inversion to these delta functions, we obtain the elementary solutions we picked earlier.\nBut from the higher point of view, one does not pick elementary solutions, but rather considers the space of all distributions which are supported on the (degenerate) conic\nξ − f = 0\n.\nWe may as well consider the distributions supported on the conic that are given by distributions of one variable on the line\nξ = f\nplus distributions on the line\nas follows: if Φ is any test function,\nwhere\ns+\n, and\ns−\n, are distributions of one variable.\nThen Fourier inversion gives, for the boundary conditions, something very similar to what we had more concretely above (put\nΦ(ξ, f) = ei2π(xξ+tf)\n, which is clearly of polynomial growth):\nand\nNow, as before, applying the one-variable Fourier transformation in the variable x to these functions of x yields two equations in the two unknown distributions\ns±\n(which can be taken to be ordinary functions if the boundary conditions are\nL1\nor\nL2\n).\nFrom a calculational point of view, the drawback of course is that one must first calculate the Fourier transforms of the boundary conditions, then assemble the solution from these, and then calculate an inverse Fourier transform.\nClosed form formulas are rare, except when there is some geometric symmetry that can be exploited, and the numerical calculations are difficult because of the oscillatory nature of the integrals, which makes convergence slow and hard to estimate.\nFor practical calculations, other methods are often used.\nThe twentieth century has seen the extension of these methods to all linear partial differential equations with polynomial coefficients, and by extending the notion of Fourier transformation to include Fourier integral operators, some non-linear equations as well.\n\n## Fourier-transform spectroscopy\nThe Fourier transform is also used in [nuclear magnetic resonance](https://bliptext.com/articles/nuclear-magnetic-resonance) (NMR) and in other kinds of [spectroscopy](https://bliptext.com/articles/spectroscopy), e.g. infrared ([FTIR](https://bliptext.com/articles/fourier-transform-infrared-spectroscopy)).\nIn NMR an exponentially shaped free induction decay (FID) signal is acquired in the time domain and Fourier-transformed to a Lorentzian line-shape in the frequency domain.\nThe Fourier transform is also used in [magnetic resonance imaging](https://bliptext.com/articles/magnetic-resonance-imaging) (MRI) and [mass spectrometry](https://bliptext.com/articles/mass-spectrometry).\n\n## Quantum mechanics\nThe Fourier transform is useful in [quantum mechanics](https://bliptext.com/articles/quantum-mechanics) in at least two different ways.\nTo begin with, the basic conceptual structure of quantum mechanics postulates the existence of pairs of [complementary variables](https://bliptext.com/articles/complementary-variables), connected by the [Heisenberg uncertainty principle](https://bliptext.com/articles/heisenberg-uncertainty-principle).\nFor example, in one dimension, the spatial variable q of, say, a particle, can only be measured by the quantum mechanical \"[position operator](https://bliptext.com/articles/position-operator)\" at the cost of losing information about the momentum p of the particle.\nTherefore, the physical state of the particle can either be described by a function, called \"the wave function\", of q or by a function of p but not by a function of both variables.\nThe variable p is called the conjugate variable to q. In classical mechanics, the physical state of a particle (existing in one dimension, for simplicity of exposition) would be given by assigning definite values to both p and q simultaneously.\nThus, the set of all possible physical states is the two-dimensional real vector space with a p-axis and a q-axis called the [phase space](https://bliptext.com/articles/phase-space).\nIn contrast, quantum mechanics chooses a polarisation of this space in the sense that it picks a subspace of one-half the dimension, for example, the q-axis alone, but instead of considering only points, takes the set of all complex-valued \"wave functions\" on this axis.\nNevertheless, choosing the p-axis is an equally valid polarisation, yielding a different representation of the set of possible physical states of the particle.\nBoth representations of the wavefunction are related by a Fourier transform, such that\nor, equivalently,\nPhysically realisable states are\nL2\n, and so by the [Plancherel theorem](https://bliptext.com/articles/plancherel-theorem), their Fourier transforms are also\nL2\n. (Note that since q is in units of distance and p is in units of momentum, the presence of the Planck constant in the exponent makes the exponent [dimensionless](https://bliptext.com/articles/nondimensionalization), as it should be.)\nTherefore, the Fourier transform can be used to pass from one way of representing the state of the particle, by a wave function of position, to another way of representing the state of the particle: by a wave function of momentum.\nInfinitely many different polarisations are possible, and all are equally valid.\nBeing able to transform states from one representation to another by the Fourier transform is not only convenient but also the underlying reason of the Heisenberg [uncertainty principle](https://bliptext.com/articles/).\nThe other use of the Fourier transform in both quantum mechanics and [quantum field theory](https://bliptext.com/articles/quantum-field-theory) is to solve the applicable wave equation.\nIn non-relativistic quantum mechanics, [Schrödinger's equation](https://bliptext.com/articles/schr-dinger-s-equation) for a time-varying wave function in one-dimension, not subject to external forces, is\nThis is the same as the heat equation except for the presence of the imaginary unit i. Fourier methods can be used to solve this equation.\nIn the presence of a potential, given by the potential energy function\nV(x)\n, the equation becomes\nThe \"elementary solutions\", as we referred to them above, are the so-called \"stationary states\" of the particle, and Fourier's algorithm, as described above, can still be used to solve the boundary value problem of the future evolution of ψ given its values for\n. Neither of these approaches is of much practical use in quantum mechanics.\nBoundary value problems and the time-evolution of the wave function is not of much practical interest: it is the stationary states that are most important.\nIn relativistic quantum mechanics, Schrödinger's equation becomes a wave equation as was usual in classical physics, except that complex-valued waves are considered.\nA simple example, in the absence of interactions with other particles or fields, is the free one-dimensional Klein–Gordon–Schrödinger–Fock equation, this time in dimensionless units,\nThis is, from the mathematical point of view, the same as the wave equation of classical physics solved above (but with a complex-valued wave, which makes no difference in the methods).\nThis is of great use in quantum field theory: each separate Fourier component of a wave can be treated as a separate harmonic oscillator and then quantized, a procedure known as \"second quantization\".\nFourier methods have been adapted to also deal with non-trivial interactions.\nFinally, the [number operator](https://bliptext.com/articles/quantum-harmonic-oscillator) of the [quantum harmonic oscillator](https://bliptext.com/articles/quantum-harmonic-oscillator) can be interpreted, for example via the [Mehler kernel](https://bliptext.com/articles/mehler-kernel), as the [generator](https://bliptext.com/articles/symmetry-in-quantum-mechanics) of the [Fourier transform](https://bliptext.com/articles/) \\mathcal{F}.\n\n## Signal processing\nThe Fourier transform is used for the spectral analysis of time-series.\nThe subject of statistical signal processing does not, however, usually apply the Fourier transformation to the signal itself.\nEven if a real signal is indeed transient, it has been found in practice advisable to model a signal by a function (or, alternatively, a stochastic process) which is stationary in the sense that its characteristic properties are constant over all time.\nThe Fourier transform of such a function does not exist in the usual sense, and it has been found more useful for the analysis of signals to instead take the Fourier transform of its autocorrelation function.\nThe autocorrelation function R of a function f is defined by\nThis function is a function of the time-lag τ elapsing between the values of f to be correlated.\nFor most functions f that occur in practice, R is a bounded even function of the time-lag τ and for typical noisy signals it turns out to be uniformly continuous with a maximum at\n.\nThe autocorrelation function, more properly called the autocovariance function unless it is normalized in some appropriate fashion, measures the strength of the correlation between the values of f separated by a time lag.\nThis is a way of searching for the correlation of f with its own past.\nIt is useful even for other statistical tasks besides the analysis of signals.\nFor example, if\nf(t)\nrepresents the temperature at time t, one expects a strong correlation with the temperature at a time lag of 24 hours.\nIt possesses a Fourier transform,\nThis Fourier transform is called the [power spectral density](https://bliptext.com/articles/spectral-density) function of f. (Unless all periodic components are first filtered out from f, this integral will diverge, but it is easy to filter out such periodicities.)\nThe power spectrum, as indicated by this density function P, measures the amount of variance contributed to the data by the frequency ξ.\nIn electrical signals, the variance is proportional to the average power (energy per unit time), and so the power spectrum describes how much the different frequencies contribute to the average power of the signal.\nThis process is called the spectral analysis of time-series and is analogous to the usual analysis of variance of data that is not a time-series ([ANOVA](https://bliptext.com/articles/anova)).\nKnowledge of which frequencies are \"important\" in this sense is crucial for the proper design of filters and for the proper evaluation of measuring apparatuses.\nIt can also be useful for the scientific analysis of the phenomena responsible for producing the data.\nThe power spectrum of a signal can also be approximately measured directly by measuring the average power that remains in a signal after all the frequencies outside a narrow band have been filtered out.\nSpectral analysis is carried out for visual signals as well.\nThe power spectrum ignores all phase relations, which is good enough for many purposes, but for video signals other types of spectral analysis must also be employed, still using the Fourier transform as a tool.\n\n# Other notations\nOther common notations for \\hat f(\\xi) include:\nIn the sciences and engineering it is also common to make substitutions like these:\nSo the transform pair can become\nA disadvantage of the capital letter notation is when expressing a transform such as or which become the more awkward and\nIn some contexts such as particle physics, the same symbol f may be used for both for a function as well as it Fourier transform, with the two only distinguished by their [argument](https://bliptext.com/articles/argument-of-a-function) I.e. would refer to the Fourier transform because of the momentum [argument](https://bliptext.com/articles/argument-of-a-function), while would refer to the original function because of the positional [argument](https://bliptext.com/articles/argument-of-a-function).\nAlthough tildes may be used as in \\tilde{f} to indicate Fourier transforms, tildes may also be used to indicate a modification of a quantity with a more [Lorentz invariant](https://bliptext.com/articles/lorentz-invariant) form, such as, so care must be taken.\nSimilarly, \\hat f often denotes the [Hilbert transform](https://bliptext.com/articles/hilbert-transform) of f.\nThe interpretation of the complex function\nf̂(ξ)\nmay be aided by expressing it in [polar coordinate](https://bliptext.com/articles/polar-coordinate) form\nin terms of the two real functions\nA(ξ)\nand\nφ(ξ)\nwhere:\nis the [amplitude](https://bliptext.com/articles/amplitude) and\nis the [phase](https://bliptext.com/articles/phase-waves) (see [arg function](https://bliptext.com/articles/arg-mathematics)).\nThen the inverse transform can be written:\nwhich is a recombination of all the frequency components of\nf(x)\n. Each component is a complex [sinusoid](https://bliptext.com/articles/sinusoid) of the form\ne2πixξ\nwhose amplitude is\nA(ξ)\nand whose initial [phase angle](https://bliptext.com/articles/phase-waves) (at\nx = 0\n) is\nφ(ξ)\n.\nThe Fourier transform may be thought of as a mapping on function spaces.\nThis mapping is here denoted and\n(f)\nis used to denote the Fourier transform of the function f. This mapping is linear, which means that can also be seen as a linear transformation on the function space and implies that the standard notation in linear algebra of applying a linear transformation to a vector (here the function\nf\n) can be used to write\nf\ninstead of\n(f)\n. Since the result of applying the Fourier transform is again a function, we can be interested in the value of this function evaluated at the value ξ for its variable, and this is denoted either as\nf(ξ)\nor as\n( f)(ξ)\n. Notice that in the former case, it is implicitly understood that is applied first to f and then the resulting function is evaluated at ξ, not the other way around.\nIn mathematics and various applied sciences, it is often necessary to distinguish between a function f and the value of f when its variable equals x, denoted\nf(x)\n. This means that a notation like\n(f(x))\nformally can be interpreted as the Fourier transform of the values of f at x. Despite this flaw, the previous notation appears frequently, often when a particular function or a function of a particular variable is to be transformed.\nFor example,\nis sometimes used to express that the Fourier transform of a [rectangular function](https://bliptext.com/articles/rectangular-function) is a [sinc function](https://bliptext.com/articles/sinc-function), or\nis used to express the shift property of the Fourier transform.\nNotice, that the last example is only correct under the assumption that the transformed function is a function of x, not of\nx0\n.\nAs discussed above, the [characteristic function](https://bliptext.com/articles/characteristic-function-probability-theory) of a random variable is the same as the [Fourier–Stieltjes transform](https://bliptext.com/articles/) of its distribution measure, but in this context it is typical to take a different convention for the constants.\nTypically characteristic function is defined\nAs in the case of the \"non-unitary angular frequency\" convention above, the factor of 2π appears in neither the normalizing constant nor the exponent.\nUnlike any of the conventions appearing above, this convention takes the opposite sign in the exponent.\n\n# Computation methods\nThe appropriate computation method largely depends how the original mathematical function is represented and the desired form of the output function.\nIn this section we consider both functions of a continuous variable, f(x), and functions of a discrete variable (i.e. ordered pairs of x and f values).\nFor discrete-valued x, the transform integral becomes a summation of sinusoids, which is still a continuous function of frequency (\\xi or \\omega).\nWhen the sinusoids are harmonically-related (i.e. when the x-values are spaced at integer multiples of an interval), the transform is called [discrete-time Fourier transform](https://bliptext.com/articles/discrete-time-fourier-transform) (DTFT).\n\n## Discrete Fourier transforms and fast Fourier transforms\nSampling the DTFT at equally-spaced values of frequency is the most common modern method of computation.\nEfficient procedures, depending on the frequency resolution needed, are described at.\nThe [discrete Fourier transform](https://bliptext.com/articles/discrete-fourier-transform) (DFT), used there, is usually computed by a [fast Fourier transform](https://bliptext.com/articles/fast-fourier-transform) (FFT) algorithm.\n\n## Analytic integration of closed-form functions\nTables of [closed-form](https://bliptext.com/articles/closed-form-expression) Fourier transforms, such as and, are created by mathematically evaluating the Fourier analysis integral (or summation) into another [closed-form](https://bliptext.com/articles/closed-form-expression) function of frequency (\\xi or \\omega).\nWhen mathematically possible, this provides a transform for a continuum of frequency values.\nMany computer algebra systems such as [Matlab](https://bliptext.com/articles/matlab) and [Mathematica](https://bliptext.com/articles/mathematica) that are capable of [symbolic integration](https://bliptext.com/articles/symbolic-integration) are capable of computing Fourier transforms analytically.\nFor example, to compute the Fourier transform of\ncos(6πt) e−πt 2\none might enter the command integrate cos(6*pi*t) exp(−pi*t^2) exp(-i*2*pi*f*t) from -inf to inf into [Wolfram Alpha](https://bliptext.com/articles/wolfram-alpha).\n\n## Numerical integration of closed-form continuous functions\nDiscrete sampling of the Fourier transform can also be done by [numerical integration](https://bliptext.com/articles/numerical-integration) of the definition at each value of frequency for which transform is desired.\nThe numerical integration approach works on a much broader class of functions than the analytic approach.\n\n## Numerical integration of a series of ordered pairs\nIf the input function is a series of ordered pairs, numerical integration reduces to just a summation over the set of data pairs.\nThe DTFT is a common subcase of this more general situation.\n\n# Tables of important Fourier transforms\nThe following tables record some closed-form Fourier transforms.\nFor functions\nf(x)\nand\ng(x)\ndenote their Fourier transforms by\nf̂\nand\nĝ\n. Only the three most common conventions are included.\nIt may be useful to notice that entry 105 gives a relationship between the Fourier transform of a function and the original function, which can be seen as relating the Fourier transform and its inverse.\n\n## Functional relationships, one-dimensional\nThe Fourier transforms in this table may be found in or.\n\n## Square-integrable functions, one-dimensional\nThe Fourier transforms in this table may be found in, , or.\n\n## Distributions, one-dimensional\nThe Fourier transforms in this table may be found in or.\n\n## Two-dimensional functions\n\n## Formulas for general \n\nn\n\n-dimensional functions\n\n# Citations","revisionCount":1},"hasMore":false,"nextCursor":null}

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