{"history":[],"article":{"id":"5fb75899-05e3-4575-857f-f04af285e135","title":"Convolution","slug":"convolution","content":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics) (in particular, [functional analysis](https://bliptext.com/articles/functional-analysis)), **convolution** is a [mathematical operation](https://bliptext.com/articles/operation-mathematics) on two [functions](https://bliptext.com/articles/function-mathematics) (f and g) that produces a third function (f*g).\nThe term convolution refers to both the result function and to the process of computing it.\nIt is defined as the [integral](https://bliptext.com/articles/integral) of the product of the two functions after one is reflected about the y-axis and shifted.\nThe integral is evaluated for all values of shift, producing the convolution function.\nThe choice of which function is reflected and shifted before the integral does not change the integral result (see [commutativity](https://bliptext.com/articles/)).\nGraphically, it expresses how the 'shape' of one function is modified by the other.\nSome features of convolution are similar to [cross-correlation](https://bliptext.com/articles/cross-correlation): for real-valued functions, of a continuous or discrete variable, convolution differs from [cross-correlation](https://bliptext.com/articles/cross-correlation) (f \\star g) only in that either f(x) or g(x) is reflected about the y-axis in convolution; thus it is a [cross-correlation](https://bliptext.com/articles/cross-correlation) of g(-x) and f(x), or f(-x) and g(x).\nFor complex-valued functions, the cross-correlation operator is the [adjoint](https://bliptext.com/articles/hermitian-adjoint) of the convolution operator.\nConvolution has applications that include [probability](https://bliptext.com/articles/probability), [statistics](https://bliptext.com/articles/statistics), [acoustics](https://bliptext.com/articles/acoustics), [spectroscopy](https://bliptext.com/articles/spectroscopy), [signal processing](https://bliptext.com/articles/signal-processing) and [image processing](https://bliptext.com/articles/image-processing), [geophysics](https://bliptext.com/articles/geo[physics](https://bliptext.com/articles/physics)), [engineering](https://bliptext.com/articles/engineering), [physics](https://bliptext.com/articles/physics), [computer vision](https://bliptext.com/articles/computer-vision) and [differential equations](https://bliptext.com/articles/differential-equation).\nThe convolution can be defined for functions on [Euclidean space](https://bliptext.com/articles/euclidean-space) and other [groups](https://bliptext.com/articles/group-mathematics) (as [algebraic structures](https://bliptext.com/articles/algebraic-structure)).\nFor example, [periodic functions](https://bliptext.com/articles/periodic-function), such as the [discrete-time Fourier transform](https://bliptext.com/articles/discrete-time-fourier-transform), can be defined on a [circle](https://bliptext.com/articles/circle) and convolved by [periodic convolution](https://bliptext.com/articles/periodic-convolution).\n(See row 18 at .) A discrete convolution can be defined for functions on the set of [integers](https://bliptext.com/articles/integer).\nGeneralizations of convolution have applications in the field of [numerical analysis](https://bliptext.com/articles/numerical-analysis) and [numerical linear algebra](https://bliptext.com/articles/numerical-linear-algebra), and in the design and implementation of [finite impulse response](https://bliptext.com/articles/finite-impulse-response) filters in signal processing.\nComputing the [inverse](https://bliptext.com/articles/inverse-function) of the convolution operation is known as [deconvolution](https://bliptext.com/articles/deconvolution).\n# [Visual comparison of convolution, cross-correlation, and autocorrelation. For the operations involving function f, and assuming the height of f is 1.0, the value of the result at 5 different points is indicated by the shaded area below each point. The symmetry of f is the reason f \\star g and g*f are identical in this example. | upload.wikimedia.org/wikipedia/commons/b/b1/comparison///convolution///correlation.svg]\n\n# Definition\nThe convolution of f and g is written f * g, denoting the operator with the symbol *.\nIt is defined as the integral of the product of the two functions after one is reflected about the y-axis and shifted.\nAs such, it is a particular kind of [integral transform](https://bliptext.com/articles/integral-transform):\nAn equivalent definition is (see [commutativity](https://bliptext.com/articles/)):\nWhile the symbol t is used above, it need not represent the time domain.\nAt each t, the convolution formula can be described as the area under the function f(\\tau) weighted by the function g(-\\tau) shifted by the amount t. As t changes, the weighting function g(t-\\tau) emphasizes different parts of the input function f(\\tau); If t is a positive value, then g(t-\\tau) is equal to g(-\\tau) that slides or is shifted along the \\tau-axis toward the right (toward +\\infty) by the amount of t, while if t is a negative value, then g(t-\\tau) is equal to g(-\\tau) that slides or is shifted toward the left (toward -\\infty) by the amount of |t|.\nFor functions f, g [supported](https://bliptext.com/articles/support-mathematics) on only [0,\\infty) (i.e., zero for negative arguments), the integration limits can be truncated, resulting in:\nFor the multi-dimensional formulation of convolution, see [domain of definition](https://bliptext.com/articles/) (below).\n\n## Notation\nA common engineering notational convention is:\nwhich has to be interpreted carefully to avoid confusion.\nFor instance, is equivalent to, but is in fact equivalent to.\n\n## Relations with other transforms\nGiven two functions f(t) and g(t) with [bilateral Laplace transforms](https://bliptext.com/articles/two-sided-laplace-transform) (two-sided Laplace transform)\nand\nrespectively, the convolution operation (f * g)(t) can be defined as the [inverse Laplace transform](https://bliptext.com/articles/inverse-laplace-transform) of the product of F(s) and G(s).\nMore precisely,\nLet t = u + v, then\nNote that is the bilateral Laplace transform of (f * g)(t).\nA similar derivation can be done using the [unilateral Laplace transform](https://bliptext.com/articles/laplace-transform) (one-sided Laplace transform).\nThe convolution operation also describes the output (in terms of the input) of an important class of operations known as linear time-invariant (LTI).\nSee [LTI system theory](https://bliptext.com/articles/lti-system-theory) for a derivation of convolution as the result of LTI constraints.\nIn terms of the [Fourier transforms](https://bliptext.com/articles/fourier-transform) of the input and output of an LTI operation, no new frequency components are created.\nThe existing ones are only modified (amplitude and/or phase).\nIn other words, the output transform is the pointwise product of the input transform with a third transform (known as a [transfer function](https://bliptext.com/articles/transfer-function)).\nSee [Convolution theorem](https://bliptext.com/articles/convolution-theorem) for a derivation of that property of convolution.\nConversely, convolution can be derived as the inverse Fourier transform of the pointwise product of two Fourier transforms.\n\n# Visual explanation\n\n# Historical developments\nOne of the earliest uses of the convolution integral appeared in [D'Alembert](https://bliptext.com/articles/jean-le-rond-d-alembert)'s derivation of [Taylor's theorem](https://bliptext.com/articles/taylor-s-theorem) in Recherches sur différents points importants du système du monde, published in 1754.\nAlso, an expression of the type:\nis used by [Sylvestre François Lacroix](https://bliptext.com/articles/sylvestre-fran-ois-lacroix) on page 505 of his book entitled Treatise on differences and series, which is the last of 3 volumes of the encyclopedic series: Traité du calcul différentiel et du calcul intégral, Chez Courcier, Paris, 1797–1800.\nSoon thereafter, convolution operations appear in the works of [Pierre Simon Laplace](https://bliptext.com/articles/pierre-simon-laplace), [Jean-Baptiste Joseph Fourier](https://bliptext.com/articles/jean-baptiste-joseph-fourier), [Siméon Denis Poisson](https://bliptext.com/articles/sim-on-denis-poisson), and others.\nThe term itself did not come into wide use until the 1950s or 1960s.\nPrior to that it was sometimes known as Faltung (which means folding in [German](https://bliptext.com/articles/german-language)), composition product, superposition integral, and [Carson](https://bliptext.com/articles/john-renshaw-carson)'s integral.\nYet it appears as early as 1903, though the definition is rather unfamiliar in older uses.\nThe operation:\nis a particular case of composition products considered by the Italian mathematician [Vito Volterra](https://bliptext.com/articles/vito-volterra) in 1913.\n\n# Circular convolution\nWhen a function g_T is periodic, with period T, then for functions, f, such that f * g_T exists, the convolution is also periodic and identical to:\nwhere t_0 is an arbitrary choice.\nThe summation is called a [periodic summation](https://bliptext.com/articles/periodic-summation) of the function f.\nWhen g_T is a periodic summation of another function, g, then f*g_T is known as a circular or cyclic convolution of f and g.\nAnd if the periodic summation above is replaced by f_T, the operation is called a periodic convolution of f_T and g_T.\n\n# Discrete convolution\nFor complex-valued functions f and g defined on the set \\Z of integers, the discrete convolution of f and g is given by:\nor equivalently (see [commutativity](https://bliptext.com/articles/)) by:\nThe convolution of two finite sequences is defined by extending the sequences to finitely supported functions on the set of integers.\nWhen the sequences are the coefficients of two [polynomials](https://bliptext.com/articles/polynomial), then the coefficients of the ordinary product of the two [polynomials](https://bliptext.com/articles/polynomial) are the convolution of the original two sequences.\nThis is known as the [Cauchy product](https://bliptext.com/articles/cauchy-product) of the coefficients of the sequences.\nThus when g has finite support in the set (representing, for instance, a [finite impulse response](https://bliptext.com/articles/finite-impulse-response)), a finite summation may be used:\n\n## Circular discrete convolution\nWhen a function g_{_N} is periodic, with period N, then for functions, f, such that f*g_{_N} exists, the convolution is also periodic and identical to**:**\nThe summation on k is called a [periodic summation](https://bliptext.com/articles/periodic-summation) of the function f.\nIf g_{_N} is a periodic summation of another function, g, then f*g_{_N} is known as a [circular convolution](https://bliptext.com/articles/circular-convolution) of f and g.\nWhen the non-zero durations of both f and g are limited to the interval [0,N-1], f*g_{_N} reduces to these common forms**:**\nThe notation f *_N g for cyclic convolution denotes convolution over the [cyclic group](https://bliptext.com/articles/cyclic-group) of [[modular arithmetic|integers modulo\nN\n]].\nCircular convolution arises most often in the context of fast convolution with a [fast Fourier transform](https://bliptext.com/articles/fast-fourier-transform) (FFT) algorithm.\n\n## Fast convolution algorithms\nIn many situations, discrete convolutions can be converted to circular convolutions so that fast transforms with a convolution property can be used to implement the computation.\nFor example, convolution of digit sequences is the kernel operation in [multiplication](https://bliptext.com/articles/multiplication) of multi-digit numbers, which can therefore be efficiently implemented with transform techniques.\nrequires N arithmetic operations per output value and\nN2\noperations for N outputs.\nThat can be significantly reduced with any of several fast algorithms.\n[Digital signal processing](https://bliptext.com/articles/digital-signal-processing) and other applications typically use fast convolution algorithms to reduce the cost of the convolution to O(N log N) complexity.\nThe most common fast convolution algorithms use [fast Fourier transform](https://bliptext.com/articles/fast-fourier-transform) (FFT) algorithms via the [circular convolution theorem](https://bliptext.com/articles/discrete-fourier-transform).\nSpecifically, the [circular convolution](https://bliptext.com/articles/circular-convolution) of two finite-length sequences is found by taking an FFT of each sequence, multiplying pointwise, and then performing an inverse FFT.\nConvolutions of the type defined above are then efficiently implemented using that technique in conjunction with zero-extension and/or discarding portions of the output.\nOther fast convolution algorithms, such as the [Schönhage–Strassen algorithm](https://bliptext.com/articles/sch-nhage-strassen-algorithm) or the Mersenne transform, use fast Fourier transforms in other [rings](https://bliptext.com/articles/ring-mathematics).\nThe Winograd method is used as an alternative to the FFT.\nIt significantly speeds up 1D, 2D, and 3D convolution.\nIf one sequence is much longer than the other, zero-extension of the shorter sequence and fast circular convolution is not the most computationally efficient method available.\nInstead, decomposing the longer sequence into blocks and convolving each block allows for faster algorithms such as the [overlap–save method](https://bliptext.com/articles/overlap-save-method) and [overlap–add method](https://bliptext.com/articles/overlap-add-method).\nA hybrid convolution method that combines block and [FIR](https://bliptext.com/articles/finite-impulse-response) algorithms allows for a zero input-output latency that is useful for real-time convolution computations.\n\n# Domain of definition\nThe convolution of two complex-valued functions on\nRd\nis itself a complex-valued function on\nRd\n, defined by:\nand is well-defined only if f and g decay sufficiently rapidly at infinity in order for the integral to exist.\nConditions for the existence of the convolution may be tricky, since a blow-up in g at infinity can be easily offset by sufficiently rapid decay in f. The question of existence thus may involve different conditions on f and g:\n\n## Compactly supported functions\nIf f and g are [compactly supported](https://bliptext.com/articles/compact-support) [continuous functions](https://bliptext.com/articles/continuous-function), then their convolution exists, and is also [compactly supported](https://bliptext.com/articles/compact-support) and continuous.\nMore generally, if either function (say f) is compactly supported and the other is [locally integrable](https://bliptext.com/articles/locally-integrable-function), then the convolution\nf∗g\nis well-defined and continuous.\nConvolution of f and g is also well defined when both functions are locally square integrable on\nR\nand supported on an interval of the form\n[a, +∞)\n(or both supported on\n[−∞, a]\n).\n\n## Integrable functions\nThe convolution of f and g exists if f and g are both [Lebesgue integrable functions](https://bliptext.com/articles/lebesgue-integral) in [[Lp space|\nL1\n(\nRd\n)]], and in this case\nf∗g\nis also integrable.\nThis is a consequence of [Tonelli's theorem](https://bliptext.com/articles/fubini-s-theorem).\nThis is also true for functions in\nL1\n, under the discrete convolution, or more generally for the [convolution on any group](https://bliptext.com/articles/).\nLikewise, if\nf ∈ L1\n(\nRd\n) and\ng ∈ Lp\n(\nRd\n) where\n1 ≤ p ≤ ∞\n, then\nf*g ∈ Lp\n(\nRd\n), and\nIn the particular case\n, this shows that\nL1\nis a [Banach algebra](https://bliptext.com/articles/banach-algebra) under the convolution (and equality of the two sides holds if f and g are non-negative almost everywhere).\nMore generally, [Young's inequality](https://bliptext.com/articles/young-s-convolution-inequality) implies that the convolution is a continuous bilinear map between suitable\nLp\nspaces.\nSpecifically, if\n1 ≤ p, q, r ≤ ∞\nsatisfy:\nthen\nso that the convolution is a continuous bilinear mapping from\nLp×Lq\nto\nLr\n. The Young inequality for convolution is also true in other contexts (circle group, convolution on\nZ\n). The preceding inequality is not sharp on the real line: when\n1 < p, q, r < ∞\n, there exists a constant\nBp,q < 1\nsuch that:\nThe optimal value of\nBp,q\nwas discovered in 1975 and independently in 1976, see [Brascamp–Lieb inequality](https://bliptext.com/articles/brascamp-lieb-inequality).\nA stronger estimate is true provided\n1 < p, q, r < ∞\nwhere \\|g\\|_{q,w} is the [[Lp space#Weak Lp|weak\nLq\n]] norm.\nConvolution also defines a bilinear continuous map for, owing to the weak Young inequality:\n\n## Functions of rapid decay\nIn addition to compactly supported functions and integrable functions, functions that have sufficiently rapid decay at infinity can also be convolved.\nAn important feature of the convolution is that if f and g both decay rapidly, then f∗g also decays rapidly.\nIn particular, if f and g are [rapidly decreasing functions](https://bliptext.com/articles/rapidly-decreasing-function), then so is the convolution f∗g.\n[](https://bliptext.com/articles/)C[](https://bliptext.com/articles/)o[](https://bliptext.com/articles/)m[](https://bliptext.com/articles/)b[](https://bliptext.com/articles/)i[](https://bliptext.com/articles/)n[](https://bliptext.com/articles/)e[](https://bliptext.com/articles/)d[](https://bliptext.com/articles/) [](https://bliptext.com/articles/)w[](https://bliptext.com/articles/)i[](https://bliptext.com/articles/)t[](https://bliptext.com/articles/)h[](https://bliptext.com/articles/) [](https://bliptext.com/articles/)t[](https://bliptext.com/articles/)h[](https://bliptext.com/articles/)e[](https://bliptext.com/articles/) [](https://bliptext.com/articles/)f[](https://bliptext.com/articles/)a[](https://bliptext.com/articles/)c[](https://bliptext.com/articles/)t[](https://bliptext.com/articles/) [](https://bliptext.com/articles/)t[](https://bliptext.com/articles/)h[](https://bliptext.com/articles/)a[](https://bliptext.com/articles/)t[](https://bliptext.com/articles/) 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[](https://bliptext.com/articles/)c[](https://bliptext.com/articles/)l[](https://bliptext.com/articles/)o[](https://bliptext.com/articles/)s[](https://bliptext.com/articles/)e[](https://bliptext.com/articles/)d[](https://bliptext.com/articles/) [](https://bliptext.com/articles/)u[](https://bliptext.com/articles/)n[](https://bliptext.com/articles/)d[](https://bliptext.com/articles/)e[](https://bliptext.com/articles/)r[](https://bliptext.com/articles/) [](https://bliptext.com/articles/)c[](https://bliptext.com/articles/)o[](https://bliptext.com/articles/)n[](https://bliptext.com/articles/)v[](https://bliptext.com/articles/)o[](https://bliptext.com/articles/)l[](https://bliptext.com/articles/)u[](https://bliptext.com/articles/)t[](https://bliptext.com/articles/)i[](https://bliptext.com/articles/)o[](https://bliptext.com/articles/)n[](https://bliptext.com/articles/).[](https://bliptext.com/articles/)\n\n## Distributions\nIf f is a smooth function that is [compactly supported](https://bliptext.com/articles/support-mathematics) and g is a distribution, then f∗g is a smooth function defined by\nMore generally, it is possible to extend the definition of the convolution in a unique way with \\varphi the same as f above, so that the associative law\nremains valid in the case where f is a distribution, and g a compactly supported distribution.\n\n## Measures\nThe convolution of any two [Borel measures](https://bliptext.com/articles/borel-measure) μ and ν of [bounded variation](https://bliptext.com/articles/bounded-variation) is the measure \\mu*\\nu defined by\nIn particular,\nwhere is a measurable set and 1_A is the [indicator function](https://bliptext.com/articles/indicator-function) of A.\nThis agrees with the convolution defined above when μ and ν are regarded as distributions, as well as the convolution of L1 functions when μ and ν are absolutely continuous with respect to the Lebesgue measure.\nThe convolution of measures also satisfies the following version of Young's inequality\nwhere the norm is the [total variation](https://bliptext.com/articles/total-variation) of a measure.\nBecause the space of measures of bounded variation is a [Banach space](https://bliptext.com/articles/banach-space), convolution of measures can be treated with standard methods of [functional analysis](https://bliptext.com/articles/functional-analysis) that may not apply for the convolution of distributions.\n\n# Properties\n\n## Algebraic properties\nThe convolution defines a product on the [linear space](https://bliptext.com/articles/linear-space) of integrable functions.\nThis product satisfies the following algebraic properties, which formally mean that the space of integrable functions with the product given by convolution is a commutative [associative algebra](https://bliptext.com/articles/associative-algebra) without [identity](https://bliptext.com/articles/identity-element).\nOther linear spaces of functions, such as the space of continuous functions of compact support, are [closed](https://bliptext.com/articles/closure-mathematics) under the convolution, and so also form commutative associative algebras.\nProof (using [convolution theorem](https://bliptext.com/articles/convolution-theorem)):\n\n## Integration\nIf f and g are integrable functions, then the integral of their convolution on the whole space is simply obtained as the product of their integrals:\nThis follows from [Fubini's theorem](https://bliptext.com/articles/fubini-s-theorem).\nThe same result holds if f and g are only assumed to be nonnegative measurable functions, by [Tonelli's theorem](https://bliptext.com/articles/fubini-s-theorem).\n\n## Differentiation\nIn the one-variable case,\nwhere is the [derivative](https://bliptext.com/articles/derivative).\nMore generally, in the case of functions of several variables, an analogous formula holds with the [partial derivative](https://bliptext.com/articles/partial-derivative):\nA particular consequence of this is that the convolution can be viewed as a \"smoothing\" operation: the convolution of f and g is differentiable as many times as f and g are in total.\nThese identities hold for example under the condition that f and g are absolutely integrable and at least one of them has an absolutely integrable (L1) weak derivative, as a consequence of [Young's convolution inequality](https://bliptext.com/articles/young-s-convolution-inequality).\nFor instance, when f is continuously differentiable with compact support, and g is an arbitrary locally integrable function,\nThese identities also hold much more broadly in the sense of tempered distributions if one of f or g is a [rapidly decreasing tempered distribution](https://bliptext.com/articles/distribution-mathematics), a compactly supported tempered distribution or a Schwartz function and the other is a tempered distribution.\nOn the other hand, two positive integrable and infinitely differentiable functions may have a nowhere continuous convolution.\nIn the discrete case, the [difference operator](https://bliptext.com/articles/difference-operator) D f(n) = f(n + 1) − f(n) satisfies an analogous relationship:\n\n## Convolution theorem\nThe [convolution theorem](https://bliptext.com/articles/convolution-theorem) states that\nwhere denotes the [Fourier transform](https://bliptext.com/articles/fourier-transform) of f.\n\n### Convolution in other types of transformations\nVersions of this theorem also hold for the [Laplace transform](https://bliptext.com/articles/laplace-transform), two-sided [Laplace transform](https://bliptext.com/articles/laplace-transform), [Z-transform](https://bliptext.com/articles/z-transform) and [Mellin transform](https://bliptext.com/articles/mellin-transform).\n\n### Convolution on matrices\nIf \\mathcal W is the [Fourier transform matrix](https://bliptext.com/articles/dft-matrix), then\nwhere \\bull is [face-splitting product](https://bliptext.com/articles/khatri-rao-product), \\otimes denotes [Kronecker product](https://bliptext.com/articles/kronecker-product), \\circ denotes [Hadamard product](https://bliptext.com/articles/hadamard-product-matrices) (this result is an evolving of [count sketch](https://bliptext.com/articles/count-sketch) properties ).\nThis can be generalized for appropriate matrices :\nfrom the properties of the [face-splitting product](https://bliptext.com/articles/face-splitting-product).\n\n## Translational equivariance\nThe convolution commutes with translations, meaning that\nwhere τxf is the translation of the function f by x defined by\nIf f is a [Schwartz function](https://bliptext.com/articles/schwartz-function), then τxf is the convolution with a translated Dirac delta function τxf = f ∗ τx δ.\nSo translation invariance of the convolution of Schwartz functions is a consequence of the associativity of convolution.\nFurthermore, under certain conditions, convolution is the most general translation invariant operation.\nInformally speaking, the following holds\nThus some translation invariant operations can be represented as convolution.\nConvolutions play an important role in the study of [time-invariant systems](https://bliptext.com/articles/time-invariant-system), and especially [LTI system theory](https://bliptext.com/articles/lti-system-theory).\nThe representing function gS is the [impulse response](https://bliptext.com/articles/impulse-response) of the transformation S.\nA more precise version of the theorem quoted above requires specifying the class of functions on which the convolution is defined, and also requires assuming in addition that S must be a [continuous linear operator](https://bliptext.com/articles/continuous-linear-operator) with respect to the appropriate [topology](https://bliptext.com/articles/topology).\nIt is known, for instance, that every continuous translation invariant continuous linear operator on L1 is the convolution with a finite [Borel measure](https://bliptext.com/articles/borel-measure).\nMore generally, every continuous translation invariant continuous linear operator on Lp for 1 ≤ p < ∞ is the convolution with a [tempered distribution](https://bliptext.com/articles/distribution-mathematics) whose [Fourier transform](https://bliptext.com/articles/fourier-transform) is bounded.\nTo wit, they are all given by bounded [Fourier multipliers](https://bliptext.com/articles/fourier-multiplier).\n\n# Convolutions on groups\nIf G is a suitable [group](https://bliptext.com/articles/group-mathematics) endowed with a [measure](https://bliptext.com/articles/measure-mathematics) λ, and if f and g are real or complex valued [integrable](https://bliptext.com/articles/lebesgue-integral) functions on G, then we can define their convolution by\nIt is not commutative in general.\nIn typical cases of interest G is a [locally compact](https://bliptext.com/articles/locally-compact) [Hausdorff](https://bliptext.com/articles/hausdorff-space) [topological group](https://bliptext.com/articles/topological-group) and λ is a (left-) [Haar measure](https://bliptext.com/articles/haar-measure).\nIn that case, unless G is [unimodular](https://bliptext.com/articles/unimodular-group), the convolution defined in this way is not the same as.\nThe preference of one over the other is made so that convolution with a fixed function g commutes with left translation in the group:\nFurthermore, the convention is also required for consistency with the definition of the convolution of measures given below.\nHowever, with a right instead of a left Haar measure, the latter integral is preferred over the former.\nOn [locally compact abelian groups](https://bliptext.com/articles/locally-compact-abelian-group), a version of the [convolution theorem](https://bliptext.com/articles/convolution-theorem) holds: the Fourier transform of a convolution is the pointwise product of the Fourier transforms.\n**T**he [circle group](https://bliptext.com/articles/circle-group) **T** with the Lebesgue measure is an immediate example.\nFor a fixed g in L1(****T****), we have the following familiar operator acting on the [Hilbert space](https://bliptext.com/articles/hilbert-space) L2(****T****):\nThe operator T is [compact](https://bliptext.com/articles/compact-operator-on-hilbert-space).\nA direct calculation shows that its adjoint T* is convolution with\nBy the commutativity property cited above, T is [normal](https://bliptext.com/articles/normal-operator): T* T = TT*.\nAlso, T commutes with the translation operators.\nConsider the family S of operators consisting of all such convolutions and the translation operators.\nThen S is a commuting family of normal operators.\nAccording to [spectral theory](https://bliptext.com/articles/compact-operator-on-hilbert-space), there exists an orthonormal basis {hk} that simultaneously diagonalizes S.\nThis characterizes convolutions on the circle.\nSpecifically, we have\nwhich are precisely the [characters](https://bliptext.com/articles/character-mathematics) of **T**.\nEach convolution is a compact [multiplication operator](https://bliptext.com/articles/multiplication-operator) in this basis.\nThis can be viewed as a version of the convolution theorem discussed above.\nA discrete example is a finite [cyclic group](https://bliptext.com/articles/cyclic-group) of order n.\nConvolution operators are here represented by [circulant matrices](https://bliptext.com/articles/circulant-matrices), and can be diagonalized by the [discrete Fourier transform](https://bliptext.com/articles/discrete-fourier-transform).\nA similar result holds for compact groups (not necessarily abelian): the matrix coefficients of finite-dimensional [unitary representations](https://bliptext.com/articles/unitary-representation) form an orthonormal basis in L2 by the [Peter–Weyl theorem](https://bliptext.com/articles/peter-weyl-theorem), and an analog of the convolution theorem continues to hold, along with many other aspects of [harmonic analysis](https://bliptext.com/articles/harmonic-analysis) that depend on the Fourier transform.\n\n# Convolution of measures\nLet G be a (multiplicatively written) topological group.\nIf μ and ν are finite [Borel measures](https://bliptext.com/articles/borel-measure) on G, then their convolution μ∗ν is defined as the [pushforward measure](https://bliptext.com/articles/pushforward-measure) of the [group action](https://bliptext.com/articles/group-action-mathematics) and can be written as\nfor each measurable subset E of G.\nThe convolution is also a finite measure, whose [total variation](https://bliptext.com/articles/total-variation) satisfies\nIn the case when G is [locally compact](https://bliptext.com/articles/locally-compact) with (left-)[Haar measure](https://bliptext.com/articles/haar-measure) λ, and μ and ν are [absolutely continuous](https://bliptext.com/articles/absolute-continuity) with respect to a λ, [so that each has a density function](https://bliptext.com/articles/radon-nikodym-theorem), then the convolution μ∗ν is also [absolutely continuous](https://bliptext.com/articles/absolute-continuity), and its density function is just the convolution of the two separate density functions.\nIf μ and ν are [probability measures](https://bliptext.com/articles/probability-measure) on the topological group (R,+), then the convolution μ∗ν is the [probability distribution](https://bliptext.com/articles/probability-distribution) of the sum X + Y of two [independent](https://bliptext.com/articles/statistical-independence) [random variables](https://bliptext.com/articles/random-variable) X and Y whose respective distributions are μ and ν.\n\n# Infimal convolution\nIn [convex analysis](https://bliptext.com/articles/convex-analysis), the **infimal convolution** of proper (not identically +\\infty) [convex functions](https://bliptext.com/articles/convex-function) on \\mathbb R^n is defined by:\nIt can be shown that the infimal convolution of convex functions is convex.\nFurthermore, it satisfies an identity analogous to that of the Fourier transform of a traditional convolution, with the role of the Fourier transform is played instead by the [Legendre transform](https://bliptext.com/articles/legendre-transform):\nWe have:\n\n# Bialgebras\nLet (X, Δ, ∇, ε, η) be a [bialgebra](https://bliptext.com/articles/bialgebra) with comultiplication Δ, multiplication ∇, unit η, and counit ε.\nThe convolution is a product defined on the [endomorphism algebra](https://bliptext.com/articles/endomorphism-algebra) End(X) as follows.\nLet φ, ψ ∈ End(X), that is, φ, ψ: X → X are functions that respect all algebraic structure of X, then the convolution φ∗ψ is defined as the composition\nThe convolution appears notably in the definition of [Hopf algebras](https://bliptext.com/articles/hopf-algebra).\nA bialgebra is a Hopf algebra if and only if it has an antipode: an endomorphism S such that\n\n# Applications\nConvolution and related operations are found in many applications in science, engineering and mathematics.\n# [Gaussian blur can be used to obtain a smooth grayscale digital image of a halftone print. | upload.wikimedia.org/wikipedia/commons/d/d7/Halftone%2C///Gaussian///Blur.jpg]","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T16:32:56.036Z","updated_at":"2025-01-25T16:32:56.036Z","search_vector":"'convolut':1","search_content":" \n \n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics) (in particular, [functional analysis](https://bliptext.com/articles/functional-analysis)), **convolution** is a [mathematical operation](https://bliptext.com/articles/operation-mathematics) on two [functions](https://bliptext.com/articles/function-mathematics) (f and g) that produces a third function (f*g).\nThe term convolution refers to both the result function and to the process of computing it.\nIt is defined as the [integral](https://bliptext.com/articles/integral) of the product of the two functions after one is reflected about the y-axis and shifted.\nThe integral is evaluated for all values of shift, producing the convolution function.\nThe choice of which function is reflected and shifted before the integral does not change the integral result (see [commutativity](https://bliptext.com/articles/)).\nGraphically, it expresses how the 'shape' of one function is modified by the other.\nSome features of convolution are similar to [cross-correlation](https://bliptext.com/articles/cross-correlation): for real-valued functions, of a continuous or discrete variable, convolution differs from [cross-correlation](https://bliptext.com/articles/cross-correlation) (f \\star g) only in that either f(x) or g(x) is reflected about the y-axis in convolution; thus it is a [cross-correlation](https://bliptext.com/articles/cross-correlation) of g(-x) and f(x), or f(-x) and g(x).\nFor complex-valued functions, the cross-correlation operator is the [adjoint](https://bliptext.com/articles/hermitian-adjoint) of the convolution operator.\nConvolution has applications that include [probability](https://bliptext.com/articles/probability), [statistics](https://bliptext.com/articles/statistics), [acoustics](https://bliptext.com/articles/acoustics), [spectroscopy](https://bliptext.com/articles/spectroscopy), [signal processing](https://bliptext.com/articles/signal-processing) and [image processing](https://bliptext.com/articles/image-processing), [geo[physics](https://bliptext.com/articles/physics)](https://bliptext.com/articles/geo[physics](https://bliptext.com/articles/physics)), [engineering](https://bliptext.com/articles/engineering), [physics](https://bliptext.com/articles/physics), [computer vision](https://bliptext.com/articles/computer-vision) and [differential equations](https://bliptext.com/articles/differential-equation).\nThe convolution can be defined for functions on [Euclidean space](https://bliptext.com/articles/euclidean-space) and other [groups](https://bliptext.com/articles/group-mathematics) (as [algebraic structures](https://bliptext.com/articles/algebraic-structure)).\nFor example, [periodic functions](https://bliptext.com/articles/periodic-function), such as the [discrete-time Fourier transform](https://bliptext.com/articles/discrete-time-fourier-transform), can be defined on a [circle](https://bliptext.com/articles/circle) and convolved by [periodic convolution](https://bliptext.com/articles/periodic-convolution).\n(See row 18 at .) A discrete convolution can be defined for functions on the set of [integers](https://bliptext.com/articles/integer).\nGeneralizations of convolution have applications in the field of [numerical analysis](https://bliptext.com/articles/numerical-analysis) and [numerical linear algebra](https://bliptext.com/articles/numerical-linear-algebra), and in the design and implementation of [finite impulse response](https://bliptext.com/articles/finite-impulse-response) filters in signal processing.\nComputing the [inverse](https://bliptext.com/articles/inverse-function) of the convolution operation is known as [deconvolution](https://bliptext.com/articles/deconvolution).\n# [Visual comparison of convolution, cross-correlation, and autocorrelation. For the operations involving function f, and assuming the height of f is 1.0, the value of the result at 5 different points is indicated by the shaded area below each point. The symmetry of f is the reason f \\star g and g*f are identical in this example. | upload.wikimedia.org/wikipedia/commons/b/b1/comparison///convolution///correlation.svg]\n\n# Definition\nThe convolution of f and g is written f * g, denoting the operator with the symbol *.\nIt is defined as the integral of the product of the two functions after one is reflected about the y-axis and shifted.\nAs such, it is a particular kind of [integral transform](https://bliptext.com/articles/integral-transform):\nAn equivalent definition is (see [commutativity](https://bliptext.com/articles/)):\nWhile the symbol t is used above, it need not represent the time domain.\nAt each t, the convolution formula can be described as the area under the function f(\\tau) weighted by the function g(-\\tau) shifted by the amount t. As t changes, the weighting function g(t-\\tau) emphasizes different parts of the input function f(\\tau); If t is a positive value, then g(t-\\tau) is equal to g(-\\tau) that slides or is shifted along the \\tau-axis toward the right (toward +\\infty) by the amount of t, while if t is a negative value, then g(t-\\tau) is equal to g(-\\tau) that slides or is shifted toward the left (toward -\\infty) by the amount of |t|.\nFor functions f, g [supported](https://bliptext.com/articles/support-mathematics) on only [0,\\infty) (i.e., zero for negative arguments), the integration limits can be truncated, resulting in:\nFor the multi-dimensional formulation of convolution, see [domain of definition](https://bliptext.com/articles/) (below).\n\n## Notation\nA common engineering notational convention is:\nwhich has to be interpreted carefully to avoid confusion.\nFor instance, is equivalent to, but is in fact equivalent to.\n\n## Relations with other transforms\nGiven two functions f(t) and g(t) with [bilateral Laplace transforms](https://bliptext.com/articles/two-sided-laplace-transform) (two-sided Laplace transform)\nand\nrespectively, the convolution operation (f * g)(t) can be defined as the [inverse Laplace transform](https://bliptext.com/articles/inverse-laplace-transform) of the product of F(s) and G(s).\nMore precisely,\nLet t = u + v, then\nNote that is the bilateral Laplace transform of (f * g)(t).\nA similar derivation can be done using the [unilateral Laplace transform](https://bliptext.com/articles/laplace-transform) (one-sided Laplace transform).\nThe convolution operation also describes the output (in terms of the input) of an important class of operations known as linear time-invariant (LTI).\nSee [LTI system theory](https://bliptext.com/articles/lti-system-theory) for a derivation of convolution as the result of LTI constraints.\nIn terms of the [Fourier transforms](https://bliptext.com/articles/fourier-transform) of the input and output of an LTI operation, no new frequency components are created.\nThe existing ones are only modified (amplitude and/or phase).\nIn other words, the output transform is the pointwise product of the input transform with a third transform (known as a [transfer function](https://bliptext.com/articles/transfer-function)).\nSee [Convolution theorem](https://bliptext.com/articles/convolution-theorem) for a derivation of that property of convolution.\nConversely, convolution can be derived as the inverse Fourier transform of the pointwise product of two Fourier transforms.\n\n# Visual explanation\n\n# Historical developments\nOne of the earliest uses of the convolution integral appeared in [D'Alembert](https://bliptext.com/articles/jean-le-rond-d-alembert)'s derivation of [Taylor's theorem](https://bliptext.com/articles/taylor-s-theorem) in Recherches sur différents points importants du système du monde, published in 1754.\nAlso, an expression of the type:\nis used by [Sylvestre François Lacroix](https://bliptext.com/articles/sylvestre-fran-ois-lacroix) on page 505 of his book entitled Treatise on differences and series, which is the last of 3 volumes of the encyclopedic series: Traité du calcul différentiel et du calcul intégral, Chez Courcier, Paris, 1797–1800.\nSoon thereafter, convolution operations appear in the works of [Pierre Simon Laplace](https://bliptext.com/articles/pierre-simon-laplace), [Jean-Baptiste Joseph Fourier](https://bliptext.com/articles/jean-baptiste-joseph-fourier), [Siméon Denis Poisson](https://bliptext.com/articles/sim-on-denis-poisson), and others.\nThe term itself did not come into wide use until the 1950s or 1960s.\nPrior to that it was sometimes known as Faltung (which means folding in [German](https://bliptext.com/articles/german-language)), composition product, superposition integral, and [Carson](https://bliptext.com/articles/john-renshaw-carson)'s integral.\nYet it appears as early as 1903, though the definition is rather unfamiliar in older uses.\nThe operation:\nis a particular case of composition products considered by the Italian mathematician [Vito Volterra](https://bliptext.com/articles/vito-volterra) in 1913.\n\n# Circular convolution\nWhen a function g_T is periodic, with period T, then for functions, f, such that f * g_T exists, the convolution is also periodic and identical to:\nwhere t_0 is an arbitrary choice.\nThe summation is called a [periodic summation](https://bliptext.com/articles/periodic-summation) of the function f.\nWhen g_T is a periodic summation of another function, g, then f*g_T is known as a circular or cyclic convolution of f and g.\nAnd if the periodic summation above is replaced by f_T, the operation is called a periodic convolution of f_T and g_T.\n\n# Discrete convolution\nFor complex-valued functions f and g defined on the set \\Z of integers, the discrete convolution of f and g is given by:\nor equivalently (see [commutativity](https://bliptext.com/articles/)) by:\nThe convolution of two finite sequences is defined by extending the sequences to finitely supported functions on the set of integers.\nWhen the sequences are the coefficients of two [polynomials](https://bliptext.com/articles/polynomial), then the coefficients of the ordinary product of the two [polynomials](https://bliptext.com/articles/polynomial) are the convolution of the original two sequences.\nThis is known as the [Cauchy product](https://bliptext.com/articles/cauchy-product) of the coefficients of the sequences.\nThus when g has finite support in the set (representing, for instance, a [finite impulse response](https://bliptext.com/articles/finite-impulse-response)), a finite summation may be used:\n\n## Circular discrete convolution\nWhen a function g_{_N} is periodic, with period N, then for functions, f, such that f*g_{_N} exists, the convolution is also periodic and identical to**:**\nThe summation on k is called a [periodic summation](https://bliptext.com/articles/periodic-summation) of the function f.\nIf g_{_N} is a periodic summation of another function, g, then f*g_{_N} is known as a [circular convolution](https://bliptext.com/articles/circular-convolution) of f and g.\nWhen the non-zero durations of both f and g are limited to the interval [0,N-1], f*g_{_N} reduces to these common forms**:**\nThe notation f *_N g for cyclic convolution denotes convolution over the [cyclic group](https://bliptext.com/articles/cyclic-group) of [[modular arithmetic|integers modulo\nN\n]].\nCircular convolution arises most often in the context of fast convolution with a [fast Fourier transform](https://bliptext.com/articles/fast-fourier-transform) (FFT) algorithm.\n\n## Fast convolution algorithms\nIn many situations, discrete convolutions can be converted to circular convolutions so that fast transforms with a convolution property can be used to implement the computation.\nFor example, convolution of digit sequences is the kernel operation in [multiplication](https://bliptext.com/articles/multiplication) of multi-digit numbers, which can therefore be efficiently implemented with transform techniques.\nrequires N arithmetic operations per output value and\nN2\noperations for N outputs.\nThat can be significantly reduced with any of several fast algorithms.\n[Digital signal processing](https://bliptext.com/articles/digital-signal-processing) and other applications typically use fast convolution algorithms to reduce the cost of the convolution to O(N log N) complexity.\nThe most common fast convolution algorithms use [fast Fourier transform](https://bliptext.com/articles/fast-fourier-transform) (FFT) algorithms via the [circular convolution theorem](https://bliptext.com/articles/discrete-fourier-transform).\nSpecifically, the [circular convolution](https://bliptext.com/articles/circular-convolution) of two finite-length sequences is found by taking an FFT of each sequence, multiplying pointwise, and then performing an inverse FFT.\nConvolutions of the type defined above are then efficiently implemented using that technique in conjunction with zero-extension and/or discarding portions of the output.\nOther fast convolution algorithms, such as the [Schönhage–Strassen algorithm](https://bliptext.com/articles/sch-nhage-strassen-algorithm) or the Mersenne transform, use fast Fourier transforms in other [rings](https://bliptext.com/articles/ring-mathematics).\nThe Winograd method is used as an alternative to the FFT.\nIt significantly speeds up 1D, 2D, and 3D convolution.\nIf one sequence is much longer than the other, zero-extension of the shorter sequence and fast circular convolution is not the most computationally efficient method available.\nInstead, decomposing the longer sequence into blocks and convolving each block allows for faster algorithms such as the [overlap–save method](https://bliptext.com/articles/overlap-save-method) and [overlap–add method](https://bliptext.com/articles/overlap-add-method).\nA hybrid convolution method that combines block and [FIR](https://bliptext.com/articles/finite-impulse-response) algorithms allows for a zero input-output latency that is useful for real-time convolution computations.\n\n# Domain of definition\nThe convolution of two complex-valued functions on\nRd\nis itself a complex-valued function on\nRd\n, defined by:\nand is well-defined only if f and g decay sufficiently rapidly at infinity in order for the integral to exist.\nConditions for the existence of the convolution may be tricky, since a blow-up in g at infinity can be easily offset by sufficiently rapid decay in f. The question of existence thus may involve different conditions on f and g:\n\n## Compactly supported functions\nIf f and g are [compactly supported](https://bliptext.com/articles/compact-support) [continuous functions](https://bliptext.com/articles/continuous-function), then their convolution exists, and is also [compactly supported](https://bliptext.com/articles/compact-support) and continuous.\nMore generally, if either function (say f) is compactly supported and the other is [locally integrable](https://bliptext.com/articles/locally-integrable-function), then the convolution\nf∗g\nis well-defined and continuous.\nConvolution of f and g is also well defined when both functions are locally square integrable on\nR\nand supported on an interval of the form\n[a, +∞)\n(or both supported on\n[−∞, a]\n).\n\n## Integrable functions\nThe convolution of f and g exists if f and g are both [Lebesgue integrable functions](https://bliptext.com/articles/lebesgue-integral) in [[Lp space|\nL1\n(\nRd\n)]], and in this case\nf∗g\nis also integrable.\nThis is a consequence of [Tonelli's theorem](https://bliptext.com/articles/fubini-s-theorem).\nThis is also true for functions in\nL1\n, under the discrete convolution, or more generally for the [convolution on any group](https://bliptext.com/articles/).\nLikewise, if\nf ∈ L1\n(\nRd\n) and\ng ∈ Lp\n(\nRd\n) where\n1 ≤ p ≤ ∞\n, then\nf*g ∈ Lp\n(\nRd\n), and\nIn the particular case\n, this shows that\nL1\nis a [Banach algebra](https://bliptext.com/articles/banach-algebra) under the convolution (and equality of the two sides holds if f and g are non-negative almost everywhere).\nMore generally, [Young's inequality](https://bliptext.com/articles/young-s-convolution-inequality) implies that the convolution is a continuous bilinear map between suitable\nLp\nspaces.\nSpecifically, if\n1 ≤ p, q, r ≤ ∞\nsatisfy:\nthen\nso that the convolution is a continuous bilinear mapping from\nLp×Lq\nto\nLr\n. The Young inequality for convolution is also true in other contexts (circle group, convolution on\nZ\n). The preceding inequality is not sharp on the real line: when\n1 < p, q, r < ∞\n, there exists a constant\nBp,q < 1\nsuch that:\nThe optimal value of\nBp,q\nwas discovered in 1975 and independently in 1976, see [Brascamp–Lieb inequality](https://bliptext.com/articles/brascamp-lieb-inequality).\nA stronger estimate is true provided\n1 < p, q, r < ∞\nwhere \\|g\\|_{q,w} is the [[Lp space#Weak Lp|weak\nLq\n]] norm.\nConvolution also defines a bilinear continuous map for, owing to the weak Young inequality:\n\n## Functions of rapid decay\nIn addition to compactly supported functions and integrable functions, functions that have sufficiently rapid decay at infinity can also be convolved.\nAn important feature of the convolution is that if f and g both decay rapidly, then f∗g also decays rapidly.\nIn particular, if f and g are [rapidly decreasing functions](https://bliptext.com/articles/rapidly-decreasing-function), then so is the convolution f∗g.\n[](https://bliptext.com/articles/)C[](https://bliptext.com/articles/)o[](https://bliptext.com/articles/)m[](https://bliptext.com/articles/)b[](https://bliptext.com/articles/)i[](https://bliptext.com/a rticles/)n[](https://bliptext.com/articles/)e[](https://bliptext.com/articles/)d[](https://bliptext.com/articles/) [](https://bliptext.com/articles/)w[](https://bliptext.com/articles/)i[](https://bliptext.com/articles/)t[](https://bliptext.com/articles/)h[](https://bliptext.com/articles/) [](https://bliptext.com/articles/)t[](https://bliptext.com/articles/)h[](https://bliptext.com/articles/)e[](https://bliptext.com/articles/) [](https://bliptext.com/articles/)f[](https://bliptext.com/articles/)a[](https://bliptext.com/articles/)c[](https://bliptext.com/articles/)t[](https://bliptext.com/articles/) [](https://bliptext.com/articles/)t[](https://bliptext.com/articles/)h[](https://bliptext.com/articles/)a[](https://bliptext.com/articles/)t[](https://bliptext.com/articles/) 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[](https://bliptext.com/articles/)S[](https://bliptext.com/articles/)c[](https://bliptext.com/articles/)h[](https://bliptext.com/articles/)w[](https://bliptext.com/articles/)a[](https://bliptext.com/a rticles/)r[](https://bliptext.com/articles/)t[](https://bliptext.com/articles/)z[](https://bliptext.com/articles/) [](https://bliptext.com/articles/)f[](https://bliptext.com/articles/)u[](https://bliptext.com/articles/)n[](https://bliptext.com/articles/)c[](https://bliptext.com/articles/)t[](https://bliptext.com/a rticles/)i[](https://bliptext.com/articles/)o[](https://bliptext.com/articles/)n[](https://bliptext.com/articles/)s[](https://bliptext.com/articles/) [](https://bliptext.com/articles/)i[](https://bliptext.com/articles/)s[](https://bliptext.com/articles/) [](https://bliptext.com/articles/)c[](https://bliptext.com/articles/)l[](https://bliptext.com/articles/)o[](https://bliptext.com/articles/)s[](https://bliptext.com/articles/)e[](https://bliptext.com/a rticles/)d[](https://bliptext.com/articles/) [](https://bliptext.com/articles/)u[](https://bliptext.com/articles/)n[](https://bliptext.com/articles/)d[](https://bliptext.com/articles/)e[](https://bliptext.com/articles/)r[](https://bliptext.com/a rticles/) [](https://bliptext.com/articles/)c[](https://bliptext.com/articles/)o[](https://bliptext.com/articles/)n[](https://bliptext.com/articles/)v[](https://bliptext.com/articles/)o[](https://bliptext.com/a rticles/)l[](https://bliptext.com/articles/)u[](https://bliptext.com/articles/)t[](https://bliptext.com/articles/)i[](https://bliptext.com/articles/)o[](https://bliptext.com/articles/)n[](https://blip text.com/articles/).[](https://bliptext.com/articles/)\n\n## Distributions\nIf f is a smooth function that is [compactly supported](https://bliptext.com/articles/support-mathematics) and g is a distribution, then f∗g is a smooth function defined by\nMore generally, it is possible to extend the definition of the convolution in a unique way with \\varphi the same as f above, so that the associative law\nremains valid in the case where f is a distribution, and g a compactly supported distribution.\n\n## Measures\nThe convolution of any two [Borel measures](https://bliptext.com/articles/borel-measure) μ and ν of [bounded variation](https://bliptext.com/articles/bounded-variation) is the measure \\mu*\\nu defined by\nIn particular,\nwhere is a measurable set and 1_A is the [indicator function](https://bliptext.com/articles/indicator-function) of A.\nThis agrees with the convolution defined above when μ and ν are regarded as distributions, as well as the convolution of L1 functions when μ and ν are absolutely continuous with respect to the Lebesgue measure.\nThe convolution of measures also satisfies the following version of Young's inequality\nwhere the norm is the [total variation](https://bliptext.com/articles/total-variation) of a measure.\nBecause the space of measures of bounded variation is a [Banach space](https://bliptext.com/articles/banach-space), convolution of measures can be treated with standard methods of [functional analysis](https://bliptext.com/articles/functional-analysis) that may not apply for the convolution of distributions.\n\n# Properties\n\n## Algebraic properties\nThe convolution defines a product on the [linear space](https://bliptext.com/articles/linear-space) of integrable functions.\nThis product satisfies the following algebraic properties, which formally mean that the space of integrable functions with the product given by convolution is a commutative [associative algebra](https://bliptext.com/articles/associative-algebra) without [identity](https://bliptext.com/articles/identity-element).\nOther linear spaces of functions, such as the space of continuous functions of compact support, are [closed](https://bliptext.com/articles/closure-mathematics) under the convolution, and so also form commutative associative algebras.\nProof (using [convolution theorem](https://bliptext.com/articles/convolution-theorem)):\n\n## Integration\nIf f and g are integrable functions, then the integral of their convolution on the whole space is simply obtained as the product of their integrals:\nThis follows from [Fubini's theorem](https://bliptext.com/articles/fubini-s-theorem).\nThe same result holds if f and g are only assumed to be nonnegative measurable functions, by [Tonelli's theorem](https://bliptext.com/articles/fubini-s-theorem).\n\n## Differentiation\nIn the one-variable case,\nwhere is the [derivative](https://bliptext.com/articles/derivative).\nMore generally, in the case of functions of several variables, an analogous formula holds with the [partial derivative](https://bliptext.com/articles/partial-derivative):\nA particular consequence of this is that the convolution can be viewed as a \"smoothing\" operation: the convolution of f and g is differentiable as many times as f and g are in total.\nThese identities hold for example under the condition that f and g are absolutely integrable and at least one of them has an absolutely integrable (L1) weak derivative, as a consequence of [Young's convolution inequality](https://bliptext.com/articles/young-s-convolution-inequality).\nFor instance, when f is continuously differentiable with compact support, and g is an arbitrary locally integrable function,\nThese identities also hold much more broadly in the sense of tempered distributions if one of f or g is a [rapidly decreasing tempered distribution](https://bliptext.com/articles/distribution-mathematics), a compactly supported tempered distribution or a Schwartz function and the other is a tempered distribution.\nOn the other hand, two positive integrable and infinitely differentiable functions may have a nowhere continuous convolution.\nIn the discrete case, the [difference operator](https://bliptext.com/articles/difference-operator) D f(n) = f(n + 1) − f(n) satisfies an analogous relationship:\n\n## Convolution theorem\nThe [convolution theorem](https://bliptext.com/articles/convolution-theorem) states that\nwhere denotes the [Fourier transform](https://bliptext.com/articles/fourier-transform) of f.\n\n### Convolution in other types of transformations\nVersions of this theorem also hold for the [Laplace transform](https://bliptext.com/articles/laplace-transform), two-sided [Laplace transform](https://bliptext.com/articles/laplace-transform), [Z-transform](https://bliptext.com/articles/z-transform) and [Mellin transform](https://bliptext.com/articles/mellin-transform).\n\n### Convolution on matrices\nIf \\mathcal W is the [Fourier transform matrix](https://bliptext.com/articles/dft-matrix), then\nwhere \\bull is [face-splitting product](https://bliptext.com/articles/khatri-rao-product), \\otimes denotes [Kronecker product](https://bliptext.com/articles/kronecker-product), \\circ denotes [Hadamard product](https://bliptext.com/articles/hadamard-product-matrices) (this result is an evolving of [count sketch](https://bliptext.com/articles/count-sketch) properties ).\nThis can be generalized for appropriate matrices :\nfrom the properties of the [face-splitting product](https://bliptext.com/articles/face-splitting-product).\n\n## Translational equivariance\nThe convolution commutes with translations, meaning that\nwhere τxf is the translation of the function f by x defined by\nIf f is a [Schwartz function](https://bliptext.com/articles/schwartz-function), then τxf is the convolution with a translated Dirac delta function τxf = f ∗ τx δ.\nSo translation invariance of the convolution of Schwartz functions is a consequence of the associativity of convolution.\nFurthermore, under certain conditions, convolution is the most general translation invariant operation.\nInformally speaking, the following holds\nThus some translation invariant operations can be represented as convolution.\nConvolutions play an important role in the study of [time-invariant systems](https://bliptext.com/articles/time-invariant-system), and especially [LTI system theory](https://bliptext.com/articles/lti-system-theory).\nThe representing function gS is the [impulse response](https://bliptext.com/articles/impulse-response) of the transformation S.\nA more precise version of the theorem quoted above requires specifying the class of functions on which the convolution is defined, and also requires assuming in addition that S must be a [continuous linear operator](https://bliptext.com/articles/continuous-linear-operator) with respect to the appropriate [topology](https://bliptext.com/articles/topology).\nIt is known, for instance, that every continuous translation invariant continuous linear operator on L1 is the convolution with a finite [Borel measure](https://bliptext.com/articles/borel-measure).\nMore generally, every continuous translation invariant continuous linear operator on Lp for 1 ≤ p < ∞ is the convolution with a [tempered distribution](https://bliptext.com/articles/distribution-mathematics) whose [Fourier transform](https://bliptext.com/articles/fourier-transform) is bounded.\nTo wit, they are all given by bounded [Fourier multipliers](https://bliptext.com/articles/fourier-multiplier).\n\n# Convolutions on groups\nIf G is a suitable [group](https://bliptext.com/articles/group-mathematics) endowed with a [measure](https://bliptext.com/articles/measure-mathematics) λ, and if f and g are real or complex valued [integrable](https://bliptext.com/articles/lebesgue-integral) functions on G, then we can define their convolution by\nIt is not commutative in general.\nIn typical cases of interest G is a [locally compact](https://bliptext.com/articles/locally-compact) [Hausdorff](https://bliptext.com/articles/hausdorff-space) [topological group](https://bliptext.com/articles/topological-group) and λ is a (left-) [Haar measure](https://bliptext.com/articles/haar-measure).\nIn that case, unless G is [unimodular](https://bliptext.com/articles/unimodular-group), the convolution defined in this way is not the same as.\nThe preference of one over the other is made so that convolution with a fixed function g commutes with left translation in the group:\nFurthermore, the convention is also required for consistency with the definition of the convolution of measures given below.\nHowever, with a right instead of a left Haar measure, the latter integral is preferred over the former.\nOn [locally compact abelian groups](https://bliptext.com/articles/locally-compact-abelian-group), a version of the [convolution theorem](https://bliptext.com/articles/convolution-theorem) holds: the Fourier transform of a convolution is the pointwise product of the Fourier transforms.\n**T**he [circle group](https://bliptext.com/articles/circle-group) **T** with the Lebesgue measure is an immediate example.\nFor a fixed g in L1(****T****), we have the following familiar operator acting on the [Hilbert space](https://bliptext.com/articles/hilbert-space) L2(****T****):\nThe operator T is [compact](https://bliptext.com/articles/compact-operator-on-hilbert-space).\nA direct calculation shows that its adjoint T* is convolution with\nBy the commutativity property cited above, T is [normal](https://bliptext.com/articles/normal-operator): T* T = TT*.\nAlso, T commutes with the translation operators.\nConsider the family S of operators consisting of all such convolutions and the translation operators.\nThen S is a commuting family of normal operators.\nAccording to [spectral theory](https://bliptext.com/articles/compact-operator-on-hilbert-space), there exists an orthonormal basis {hk} that simultaneously diagonalizes S.\nThis characterizes convolutions on the circle.\nSpecifically, we have\nwhich are precisely the [characters](https://bliptext.com/articles/character-mathematics) of **T**.\nEach convolution is a compact [multiplication operator](https://bliptext.com/articles/multiplication-operator) in this basis.\nThis can be viewed as a version of the convolution theorem discussed above.\nA discrete example is a finite [cyclic group](https://bliptext.com/articles/cyclic-group) of order n.\nConvolution operators are here represented by [circulant matrices](https://bliptext.com/articles/circulant-matrices), and can be diagonalized by the [discrete Fourier transform](https://bliptext.com/articles/discrete-fourier-transform).\nA similar result holds for compact groups (not necessarily abelian): the matrix coefficients of finite-dimensional [unitary representations](https://bliptext.com/articles/unitary-representation) form an orthonormal basis in L2 by the [Peter–Weyl theorem](https://bliptext.com/articles/peter-weyl-theorem), and an analog of the convolution theorem continues to hold, along with many other aspects of [harmonic analysis](https://bliptext.com/articles/harmonic-analysis) that depend on the Fourier transform.\n\n# Convolution of measures\nLet G be a (multiplicatively written) topological group.\nIf μ and ν are finite [Borel measures](https://bliptext.com/articles/borel-measure) on G, then their convolution μ∗ν is defined as the [pushforward measure](https://bliptext.com/articles/pushforward-measure) of the [group action](https://bliptext.com/articles/group-action-mathematics) and can be written as\nfor each measurable subset E of G.\nThe convolution is also a finite measure, whose [total variation](https://bliptext.com/articles/total-variation) satisfies\nIn the case when G is [locally compact](https://bliptext.com/articles/locally-compact) with (left-)[Haar measure](https://bliptext.com/articles/haar-measure) λ, and μ and ν are [absolutely continuous](https://bliptext.com/articles/absolute-continuity) with respect to a λ, [so that each has a density function](https://bliptext.com/articles/radon-nikodym-theorem), then the convolution μ∗ν is also [absolutely continuous](https://bliptext.com/articles/absolute-continuity), and its density function is just the convolution of the two separate density functions.\nIf μ and ν are [probability measures](https://bliptext.com/articles/probability-measure) on the topological group (R,+), then the convolution μ∗ν is the [probability distribution](https://bliptext.com/articles/probability-distribution) of the sum X + Y of two [independent](https://bliptext.com/articles/statistical-independence) [random variables](https://bliptext.com/articles/random-variable) X and Y whose respective distributions are μ and ν.\n\n# Infimal convolution\nIn [convex analysis](https://bliptext.com/articles/convex-analysis), the **infimal convolution** of proper (not identically +\\infty) [convex functions](https://bliptext.com/articles/convex-function) on \\mathbb R^n is defined by:\nIt can be shown that the infimal convolution of convex functions is convex.\nFurthermore, it satisfies an identity analogous to that of the Fourier transform of a traditional convolution, with the role of the Fourier transform is played instead by the [Legendre transform](https://bliptext.com/articles/legendre-transform):\nWe have:\n\n# Bialgebras\nLet (X, Δ, ∇, ε, η) be a [bialgebra](https://bliptext.com/articles/bialgebra) with comultiplication Δ, multiplication ∇, unit η, and counit ε.\nThe convolution is a product defined on the [endomorphism algebra](https://bliptext.com/articles/endomorphism-algebra) End(X) as follows.\nLet φ, ψ ∈ End(X), that is, φ, ψ: X → X are functions that respect all algebraic structure of X, then the convolution φ∗ψ is defined as the composition\nThe convolution appears notably in the definition of [Hopf algebras](https://bliptext.com/articles/hopf-algebra).\nA bialgebra is a Hopf algebra if and only if it has an antipode: an endomorphism S such that\n\n# Applications\nConvolution and related operations are found in many applications in science, engineering and mathematics.\n# [Gaussian blur can be used to obtain a smooth grayscale digital image of a halftone print. | upload.wikimedia.org/wikipedia/commons/d/d7/Halftone%2C///Gaussian///Blur.jpg]","revisionCount":0},"hasMore":false,"nextCursor":null}

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