{"history":[],"article":{"id":"8173ea23-1b3c-4fd0-bb50-22e46597ebdd","title":"Chebyshev polynomials","slug":"chebyshev-polynomials","content":":::summary\n:::\n\n# \nThe **Chebyshev [polynomials](https://bliptext.com/articles/polynomial)** are two sequences of [polynomials](https://bliptext.com/articles/polynomial) related to the [cosine and sine functions](https://bliptext.com/articles/trigonometric-functions), notated as T_n(x) and U_n(x).\nThey can be defined in several equivalent ways, one of which starts with [trigonometric functions](https://bliptext.com/articles/trigonometric-functions):\nThe **Chebyshev polynomials of the first kind** T_n are defined by:\nSimilarly, the **Chebyshev polynomials of the second kind** U_n are defined by:\nThat these expressions define polynomials in \\cos\\theta may not be obvious at first sight but follows by rewriting and using [de Moivre's formula](https://bliptext.com/articles/de-moivre-s-formula) or by using the [angle sum formulas](https://bliptext.com/articles/list-of-trigonometric-identities) for \\cos and \\sin repeatedly.\nFor example, the [double angle formulas](https://bliptext.com/articles/list-of-trigonometric-identities), which follow directly from the angle sum formulas, may be used to obtain and, which are respectively a polynomial in \\cos\\theta and a polynomial in \\cos\\theta multiplied by \\sin\\theta.\nHence and U_1(x) = 2x.\nAn important and convenient property of the\nTn(x)\nis that they are orthogonal with respect to the following [inner product](https://bliptext.com/articles/inner-product):\nand\nUn(x)\nare orthogonal with respect to another, analogous inner product, given below.\nThe Chebyshev polynomials\nTn\nare polynomials with the largest possible leading coefficient whose [absolute value](https://bliptext.com/articles/absolute-value) on the [interval](https://bliptext.com/articles/interval-mathematics) is bounded by 1.\nThey are also the \"extremal\" polynomials for many other properties.\nIn 1952, [Cornelius Lanczos](https://bliptext.com/articles/cornelius-lanczos) showed that the Chebyshev polynomials are important in [approximation theory](https://bliptext.com/articles/approximation-theory) for the solution of linear systems; the [roots](https://bliptext.com/articles/root-of-a-polynomial) of\nTn(x)\n, which are also called [Chebyshev nodes](https://bliptext.com/articles/chebyshev-nodes), are used as matching points for optimizing [polynomial interpolation](https://bliptext.com/articles/polynomial-interpolation).\nThe resulting interpolation polynomial minimizes the problem of [Runge's phenomenon](https://bliptext.com/articles/runge-s-phenomenon) and provides an approximation that is close to the best polynomial approximation to a [continuous function](https://bliptext.com/articles/continuous-function) under the [maximum norm](https://bliptext.com/articles/maximum-norm), also called the \"[minimax](https://bliptext.com/articles/minimax)\" criterion.\nThis approximation leads directly to the method of [Clenshaw–Curtis quadrature](https://bliptext.com/articles/clenshaw-curtis-quadrature).\nThese polynomials were named after [Pafnuty Chebyshev](https://bliptext.com/articles/pafnuty-chebyshev).\nThe letter T is used because of the alternative [transliterations](https://bliptext.com/articles/transliteration) of the name Chebyshev as Tchebycheff, Tchebyshev (French) or Tschebyschow (German).\n# [Chebyshev polynomials | upload.wikimedia.org/wikipedia/commons/4/4d/Plot///of///the///Chebyshev///polynomial///of///the///first///kind///T///n%28x%29///with///n%3D5///in///the///complex///plane///from///-2-2i///to///2%2B2i///with///colors///created///with///Mathematica///13.1///function///ComplexPlot3D.svg]\n\n# Definitions\n\n## Recurrence definition\nThe **Chebyshev polynomials of the first kind** are obtained from the [recurrence relation](https://bliptext.com/articles/recurrence-relation):\nThe recurrence also allows to represent them explicitly as the determinant of a [tridiagonal matrix](https://bliptext.com/articles/tridiagonal-matrix) of size k \\times k:\nThe [ordinary generating function](https://bliptext.com/articles/generating-function) for Tn is:\nThere are several other [generating functions](https://bliptext.com/articles/generating-function) for the Chebyshev polynomials; the [exponential generating function](https://bliptext.com/articles/exponential-generating-function) is:\nThe generating function relevant for 2-dimensional [potential theory](https://bliptext.com/articles/potential-theory) and [multipole expansion](https://bliptext.com/articles/cylindrical-multipole-moments) is:\nThe **Chebyshev polynomials of the second kind** are defined by the recurrence relation:\nNotice that the two sets of recurrence relations are identical, except for T_1(x) = x vs. The ordinary generating function for Un is:\nand the exponential generating function is:\n# [Plot of the first five Tn Chebyshev polynomials (first kind) | upload.wikimedia.org/wikipedia/commons/5/5c/Chebyshev///Polynomials///of///the///First///Kind.svg]\n# [Plot of the first five Un Chebyshev polynomials (second kind) | upload.wikimedia.org/wikipedia/commons/5/53/Chebyshev///Polynomials///of///the///Second///Kind.svg]\n\n## Trigonometric definition\nAs described in the introduction, the Chebyshev polynomials of the first kind can be defined as the unique polynomials satisfying:\nor, in other words, as the unique polynomials satisfying:\nfor\nn = 0, 1, 2, 3, …\n.\nThe polynomials of the second kind satisfy:\nor\nwhich is structurally quite similar to the [Dirichlet kernel](https://bliptext.com/articles/dirichlet-kernel)\nDn(x)\n(The Dirichlet kernel, in fact, coincides with what is now known as the [Chebyshev polynomial of the fourth kind](https://bliptext.com/articles/).)\nAn equivalent way to state this is via exponentiation of a [complex number](https://bliptext.com/articles/complex-number): given a [complex number](https://bliptext.com/articles/complex-number)\nz = a + bi\nwith absolute value of one:\nChebyshev polynomials can be defined in this form when studying [trigonometric polynomials](https://bliptext.com/articles/trigonometric-polynomials).\nThat\ncos nx\nis an nth-[degree](https://bliptext.com/articles/degree-of-a-polynomial) polynomial in\ncos x\ncan be seen by observing that\ncos nx\nis the [real part](https://bliptext.com/articles/complex-number) of one side of [de Moivre's formula](https://bliptext.com/articles/de-moivre-s-formula):\nThe real part of the other side is a polynomial in\ncos x\nand\nsin x\n, in which all powers of\nsin x\nare [even](https://bliptext.com/articles/parity-mathematics) and thus replaceable through the identity\ncos2 x + sin2 x = 1\n. By the same reasoning,\nsin nx\nis the [imaginary part](https://bliptext.com/articles/complex-number) of the polynomial, in which all powers of\nsin x\nare [odd](https://bliptext.com/articles/parity-mathematics) and thus, if one factor of\nsin x\nis factored out, the remaining factors can be replaced to create a\n(n−1)\nst-degree polynomial in\ncos x\n.\n\n## Commuting polynomials definition\nChebyshev polynomials can also be characterized by the following theorem:\nIf F_n(x) is a family of monic polynomials with coefficients in a field of characteristic 0 such that and for all\nm and n, then, up to a simple change of variables, either for all n or for all n.\n\n## Pell equation definition\nThe Chebyshev polynomials can also be defined as the solutions to the [Pell equation](https://bliptext.com/articles/pell-equation):\nin a [ring](https://bliptext.com/articles/ring-mathematics)\nR[x]\n. Thus, they can be generated by the standard technique for Pell equations of taking powers of a fundamental solution:\n\n# Relations between the two kinds of Chebyshev polynomials\nThe Chebyshev polynomials of the first and second kinds correspond to a complementary pair of [Lucas sequences](https://bliptext.com/articles/lucas-sequence)\nṼn(P, Q)\nand\nŨn(P, Q)\nwith parameters\nP = 2x\nand\nQ = 1\nIt follows that they also satisfy a pair of mutual recurrence equations:\nThe second of these may be rearranged using the [recurrence definition](https://bliptext.com/articles/) for the Chebyshev polynomials of the second kind to give:\nUsing this formula iteratively gives the sum formula:\nwhile replacing U_n(x) and U_{n-2}(x) using the [derivative formula](https://bliptext.com/articles/) for T_n(x) gives the recurrence relationship for the derivative of T_n:\nThis relationship is used in the [Chebyshev spectral method](https://bliptext.com/articles/chebyshev-spectral-method) of solving differential equations.\n[Turán's inequalities](https://bliptext.com/articles/tur-n-s-inequalities) for the Chebyshev polynomials are:\nThe [integral](https://bliptext.com/articles/integral) relations are\nwhere integrals are considered as principal value.\n\n# Explicit expressions\nDifferent approaches to defining Chebyshev polynomials lead to different explicit expressions.\nThe trigonometric definition gives an explicit formula as follows:\nFrom this trigonometric form, the recurrence definition can be recovered by computing directly that the bases cases hold:\nand\nand that the [product-to-sum identity](https://bliptext.com/articles/list-of-trigonometric-identities) holds:\nUsing the complex number exponentiation definition of the Chebyshev polynomial, one can derive the following expression:\nThe two are equivalent because.\nAn explicit form of the Chebyshev polynomial in terms of monomials\nxk\nfollows from [de Moivre's formula](https://bliptext.com/articles/de-moivre-s-formula):\nwhere\nRe\ndenotes the [real part](https://bliptext.com/articles/complex-number) of a complex number.\nExpanding the formula, one gets:\nThe real part of the expression is obtained from summands corresponding to even indices.\nNoting and, one gets the explicit formula:\nwhich in turn means that:\nThis can be written as a\n2F1\n[hypergeometric function](https://bliptext.com/articles/hypergeometric-function):\nwith inverse:\nwhere the prime at the summation symbol indicates that the contribution of\nneeds to be halved if it appears.\nA related expression for\nTn\nas a sum of monomials with binomial coefficients and powers of two is\nSimilarly,\nUn\ncan be expressed in terms of hypergeometric functions:\n\n# Properties\n\n## Symmetry\nThat is, Chebyshev polynomials of even order have [even symmetry](https://bliptext.com/articles/even-and-odd-functions) and therefore contain only even powers of x. Chebyshev polynomials of odd order have [odd symmetry](https://bliptext.com/articles/even-and-odd-functions) and therefore contain only odd powers of x.\n\n## Roots and extrema\nA Chebyshev polynomial of either kind with degree n has n different [simple roots](https://bliptext.com/articles/simple-root), called **Chebyshev roots**, in the interval.\nThe roots of the Chebyshev polynomial of the first kind are sometimes called [Chebyshev nodes](https://bliptext.com/articles/chebyshev-nodes) because they are used as nodes in polynomial interpolation.\nUsing the trigonometric definition and the fact that:\none can show that the roots of Tn are:\nSimilarly, the roots of Un are:\nThe [extrema](https://bliptext.com/articles/maxima-and-minima) of Tn on the interval\n−1 ≤ x ≤ 1\nare located at:\nOne unique property of the Chebyshev polynomials of the first kind is that on the interval\n−1 ≤ x ≤ 1\nall of the [extrema](https://bliptext.com/articles/maxima-and-minima) have values that are either −1 or 1.\nThus these polynomials have only two finite [critical values](https://bliptext.com/articles/critical-value-critical-point), the defining property of [Shabat polynomials](https://bliptext.com/articles/shabat-polynomial).\nBoth the first and second kinds of Chebyshev polynomial have extrema at the endpoints, given by:\nThe [extrema](https://bliptext.com/articles/maxima-and-minima) of T_n(x) on the interval where n>0 are located at n+1 values of x. They are \\pm 1, or where d > 2, d \\;|\\; 2n, 0 < k < d/2 and (k, d) = 1, i.e., k and d are relatively prime numbers.\nSpecifically, when n is even:\nWhen n is odd:\nThis result has been generalized to solutions of, and to and for Chebyshev polynomials of the third and fourth kinds, respectively.\n\n## Differentiation and integration\nThe derivatives of the polynomials can be less than straightforward.\nBy differentiating the polynomials in their trigonometric forms, it can be shown that:\nThe last two formulas can be numerically troublesome due to the division by zero (0⁄0 [indeterminate form](https://bliptext.com/articles/indeterminate-form), specifically) at\nx = 1\nand\nx = −1\n. By [L'Hôpital's rule](https://bliptext.com/articles/l-h-pital-s-rule):\nMore generally,\nwhich is of great use in the numerical solution of [eigenvalue](https://bliptext.com/articles/eigenvalue) problems.\nAlso, we have:\nwhere the prime at the summation symbols means that the term contributed by\nk = 0\nis to be halved, if it appears.\nConcerning integration, the first derivative of the Tn implies that:\nand the recurrence relation for the first kind polynomials involving derivatives establishes that for\nn ≥ 2\nThe last formula can be further manipulated to express the integral of Tn as a function of Chebyshev polynomials of the first kind only:\nFurthermore, we have:\n\n## Products of Chebyshev polynomials\nThe Chebyshev polynomials of the first kind satisfy the relation:\nwhich is easily proved from the [product-to-sum formula](https://bliptext.com/articles/list-of-trigonometric-identities) for the cosine:\nFor\nn = 1\nthis results in the already known recurrence formula, just arranged differently, and with\nn = 2\nit forms the recurrence relation for all even or all odd indexed Chebyshev polynomials (depending on the parity of the lowest m) which implies the evenness or oddness of these polynomials.\nThree more useful formulas for evaluating Chebyshev polynomials can be concluded from this product expansion:\nThe polynomials of the second kind satisfy the similar relation:\n(with the definition\nU−1 ≡ 0\nby convention ). They also satisfy:\nfor\nm ≥ n\n. For\nn = 2\nthis recurrence reduces to:\nwhich establishes the evenness or oddness of the even or odd indexed Chebyshev polynomials of the second kind depending on whether m starts with 2 or 3.\n\n## Composition and divisibility properties\nThe trigonometric definitions of\nTn\nand\nUn\nimply the composition or nesting properties:\nFor\nTmn\nthe order of composition may be reversed, making the family of polynomial functions\nTn\na [commutative](https://bliptext.com/articles/commutative) [semigroup](https://bliptext.com/articles/semigroup) under composition.\nSince\nTm(x)\nis divisible by x if m is odd, it follows that\nTmn(x)\nis divisible by\nTn(x)\nif m is odd.\nFurthermore,\nUmn−1(x)\nis divisible by\nUn−1(x)\n, and in the case that m is even, divisible by\nTn(x)Un−1(x)\n.\n\n## Orthogonality\nBoth Tn and Un form a sequence of [orthogonal polynomials](https://bliptext.com/articles/orthogonal-polynomials).\nThe polynomials of the first kind Tn are orthogonal with respect to the weight:\non the interval, i.e. we have:\nThis can be proven by letting\nx = cos θ\nand using the defining identity\nTn(cos θ) = cos(nθ)\n.\nSimilarly, the polynomials of the second kind Un are orthogonal with respect to the weight:\non the interval, i.e. we have:\n(The measure\n√1 − x2 dx\nis, to within a normalizing constant, the [Wigner semicircle distribution](https://bliptext.com/articles/wigner-semicircle-distribution).)\nThese orthogonality properties follow from the fact that the Chebyshev polynomials solve the [Chebyshev differential equations](https://bliptext.com/articles/chebyshev-equation):\nwhich are [Sturm–Liouville differential equations](https://bliptext.com/articles/sturm-liouville-problem).\nIt is a general feature of such [differential equations](https://bliptext.com/articles/differential-equation) that there is a distinguished orthonormal set of solutions.\n(Another way to define the Chebyshev polynomials is as the solutions to [those equations](https://bliptext.com/articles/sturm-liouville-problem).)\nThe Tn also satisfy a discrete orthogonality condition:\nwhere N is any integer greater than\nmax(i, j)\n, and the\nxk\nare the N [Chebyshev nodes](https://bliptext.com/articles/chebyshev-nodes) (see above) of\nTN (x)\nFor the polynomials of the second kind and any integer\nN > i + j\nwith the same Chebyshev nodes\nxk\n, there are similar sums:\nand without the weight function:\nFor any integer\nN > i + j\n, based on the N zeros of\nUN (x)\none can get the sum:\nand again without the weight function:\n\n## Minimal \n\n∞\n\n-norm\nFor any given\nn ≥ 1\n, among the polynomials of degree n with leading coefficient 1 ([monic](https://bliptext.com/articles/monic-polynomial) polynomials):\nis the one of which the maximal absolute value on the interval is minimal.\nThis maximal absolute value is:\nand\nreaches this maximum exactly\nn + 1\ntimes at:\n\n### Remark\nBy the [equioscillation theorem](https://bliptext.com/articles/equioscillation-theorem), among all the polynomials of degree\n≤ n\n, the polynomial f minimizes\n f ∞\non [if and only if](https://bliptext.com/articles/if-and-only-if) there are\nn + 2\npoints\n−1 ≤ x0 < x1 < ⋯ < xn + 1 ≤ 1\nsuch that\n= f ∞\n.\nOf course, the null polynomial on the interval can be approximated by itself and minimizes the\n∞\n-norm.\nAbove, however,\nreaches its maximum only\nn + 1\ntimes because we are searching for the best polynomial of degree\nn ≥ 1\n(therefore the theorem evoked previously cannot be used).\n\n## Chebyshev polynomials as special cases of more general polynomial families\nThe Chebyshev polynomials are a special case of the ultraspherical or [Gegenbauer polynomials](https://bliptext.com/articles/gegenbauer-polynomials), which themselves are a special case of the [Jacobi polynomials](https://bliptext.com/articles/jacobi-polynomials) :\nChebyshev polynomials are also a special case of [Dickson polynomials](https://bliptext.com/articles/dickson-polynomial):\nIn particular, when, they are related by and.\n\n## Other properties\nThe curves given by\n, or equivalently, by the parametric equations\n,\n, are a special case of [Lissajous curves](https://bliptext.com/articles/lissajous-curve) with frequency ratio equal to n.\nSimilar to the formula:\nwe have the analogous formula:\nFor\nx ≠ 0\nand:\nwhich follows from the fact that this holds by definition for\nx = eiθ\n.\nThere are relations between Legendre polynomials and Chebyshev polynomials\nThese identities can be proven using generating functions and discrete convolution\n\n# Examples\n\n## First kind\n[[File:Chebyshev Polynomials of the 1st Kind (n=0-5, x=(-1,1)).svg|thumb|300px|The first few Chebyshev polynomials of the first kind in the domain\n−1 < x < 1\nThe flat\nT0\n,\nT1\n,\nT2\n,\nT3\n,\nT4\nand\nT5\n.]]\nThe first few Chebyshev polynomials of the first kind are\n\n## Second kind\n[[File:Chebyshev Polynomials of the 2nd Kind (n=0-5, x=(-1,1)).svg|thumb|300px|The first few Chebyshev polynomials of the second kind in the domain\n−1 < x < 1\nU0\n,\nU1\n,\nU2\n,\nU3\n,\nU4\nand\nU5\n. Although not visible in the image,\nUn(1) = n + 1\nand\nUn(−1) = (n + 1)(−1)n\n.]]\nThe first few Chebyshev polynomials of the second kind are\n\n# As a basis set\n[[Image:ChebyshevExpansion.png|thumb|right|240px|The non-smooth function (top)\ny = −x3H(−x)\n, where H is the [Heaviside step function](https://bliptext.com/articles/heaviside-step-function), and (bottom) the 5th partial sum of its Chebyshev expansion.\nThe 7th sum is indistinguishable from the original function at the resolution of the graph.]]\nIn the appropriate [Sobolev space](https://bliptext.com/articles/sobolev-space), the set of Chebyshev polynomials form an [orthonormal basis](https://bliptext.com/articles/hilbert-space), so that a function in the same space can, on\n−1 ≤ x ≤ 1\n, be expressed via the expansion:\nFurthermore, as mentioned previously, the Chebyshev polynomials form an [orthogonal](https://bliptext.com/articles/orthogonal) basis which (among other things) implies that the coefficients\nan\ncan be determined easily through the application of an [inner product](https://bliptext.com/articles/inner-product).\nThis sum is called a **Chebyshev series** or a **Chebyshev expansion**.\nSince a Chebyshev series is related to a [Fourier cosine series](https://bliptext.com/articles/fourier-cosine-series) through a change of variables, all of the theorems, identities, etc. that apply to [Fourier series](https://bliptext.com/articles/fourier-series) have a Chebyshev counterpart.\nThese attributes include:\nf(x)\nif the function is [piecewise](https://bliptext.com/articles/piecewise) [smooth](https://bliptext.com/articles/smooth-function) and [continuous](https://bliptext.com/articles/continuous-function).\nThe smoothness requirement can be relaxed in most cases – as long as there are a finite number of discontinuities in\nf(x)\nand its derivatives.\nThe abundance of the theorems and identities inherited from [Fourier series](https://bliptext.com/articles/fourier-series) make the Chebyshev polynomials important tools in [numeric analysis](https://bliptext.com/articles/numeric-analysis); for example they are the most popular general purpose basis functions used in the [spectral method](https://bliptext.com/articles/spectral-method), often in favor of trigonometric series due to generally faster convergence for continuous functions ([Gibbs' phenomenon](https://bliptext.com/articles/gibbs-phenomenon) is still a problem).\n\n## Example 1\nConsider the Chebyshev expansion of\nlog(1 + x)\n. One can express:\nOne can find the coefficients\nan\neither through the application of an inner product or by the discrete orthogonality condition.\nFor the inner product:\nwhich gives:\nAlternatively, when the inner product of the function being approximated cannot be evaluated, the discrete orthogonality condition gives an often useful result for approximate coefficients:\nwhere δij is the [Kronecker delta](https://bliptext.com/articles/kronecker-delta) function and the xk are the N Gauss–Chebyshev zeros of\nTN (x)\nFor any N, these approximate coefficients provide an exact approximation to the function at xk with a controlled error between those points.\nThe exact coefficients are obtained with\nN = ∞\n, thus representing the function exactly at all points in.\nThe rate of convergence depends on the function and its smoothness.\nThis allows us to compute the approximate coefficients an very efficiently through the [discrete cosine transform](https://bliptext.com/articles/discrete-cosine-transform):\n\n## Example 2\nTo provide another example:\n\n## Partial sums\nThe partial sums of:\nare very useful in the [approximation](https://bliptext.com/articles/approximation-theory) of various functions and in the solution of [differential equations](https://bliptext.com/articles/differential-equation) (see [spectral method](https://bliptext.com/articles/spectral-method)).\nTwo common methods for determining the coefficients an are through the use of the [inner product](https://bliptext.com/articles/inner-product) as in [Galerkin's method](https://bliptext.com/articles/galerkin-s-method) and through the use of [collocation](https://bliptext.com/articles/collocation-method) which is related to [interpolation](https://bliptext.com/articles/interpolation).\nAs an interpolant, the N coefficients of the\n(N − 1)\nst partial sum are usually obtained on the Chebyshev–Gauss–Lobatto points (or Lobatto grid), which results in minimum error and avoids [Runge's phenomenon](https://bliptext.com/articles/runge-s-phenomenon) associated with a uniform grid.\nThis collection of points corresponds to the extrema of the highest order polynomial in the sum, plus the endpoints and is given by:\n\n## Polynomial in Chebyshev form\nAn arbitrary polynomial of degree N can be written in terms of the Chebyshev polynomials of the first kind.\nSuch a polynomial\np(x)\nis of the form:\nPolynomials in Chebyshev form can be evaluated using the [Clenshaw algorithm](https://bliptext.com/articles/clenshaw-algorithm).\n\n# Families of polynomials related to Chebyshev polynomials\nPolynomials denoted C_n(x) and S_n(x) closely related to Chebyshev polynomials are sometimes used.\nThey are defined by:\nand satisfy:\nA. F. Horadam called the polynomials C_n(x) **Vieta–Lucas polynomials** and denoted them v_n(x).\nHe called the polynomials S_n(x) **Vieta–Fibonacci polynomials** and denoted them V_n(x).\nLists of both sets of polynomials are given in [Viète's](https://bliptext.com/articles/fran-ois-vi-te) Opera Mathematica, Chapter IX, Theorems VI and VII.\nThe Vieta–Lucas and Vieta–Fibonacci polynomials of real argument are, up to a power of i and a shift of index in the case of the latter, equal to [Lucas and Fibonacci polynomials](https://bliptext.com/articles/fibonacci-polynomials)\nLn\nand\nFn\nof imaginary argument.\n**Shifted Chebyshev polynomials** of the first and second kinds are related to the Chebyshev polynomials by:\nWhen the argument of the Chebyshev polynomial satisfies\n2x − 1 ∈\nthe argument of the shifted Chebyshev polynomial satisfies\nx ∈\n. Similarly, one can define shifted polynomials for generic intervals.\nAround 1990 the terms \"third-kind\" and \"fourth-kind\" came into use in connection with Chebyshev polynomials, although the polynomials denoted by these terms had an earlier development under the name **airfoil polynomials**.\nAccording to J. C. Mason and G. H. Elliott, the terminology \"third-kind\" and \"fourth-kind\" is due to [Walter Gautschi](https://bliptext.com/articles/walter-gautschi), \"in consultation with colleagues in the field of orthogonal polynomials.\"\nThe **Chebyshev polynomials of the third kind** are defined as:\nand the **Chebyshev polynomials of the fourth kind** are defined as:\nwhere.\nIn the airfoil literature V_n(x) and W_n(x) are denoted t_n(x) and u_n(x).\nThe polynomial families T_n(x), U_n(x), V_n(x), and W_n(x) are orthogonal with respect to the weights:\nand are proportional to Jacobi polynomials with:\nAll four families satisfy the recurrence with p_0(x) = 1, where p_n = T_n, U_n, V_n, or W_n, but they differ according to whether p_1(x) equals x, 2x, 2x-1, or 2x+1.\n\n## Even order modified Chebyshev polynomials\nSome applications rely on Chebyshev polynomials but may be unable to accommodate the lack of a root at zero, which rules out the use of standard Chebyshev polynomials for these kinds of applications.\nEven order Chebyshev filter designs using equally terminated passive networks are an example of this.\nHowever, even order Chebyshev polynomials may be modified to move the lowest roots down to zero while still maintaining the desirable Chebyshev equi-ripple effect.\nSuch modified polynomials contain two roots at zero, and may be referred to as even order modified Chebyshev polynomials.\nEven order modified Chebyshev polynomials may be created from the [Chebyshev nodes](https://bliptext.com/articles/chebyshev-nodes) in the same manner as standard Chebyshev polynomials.\nwhere\nIn the case of even order modified Chebyshev polynomials, the [even order modified Chebyshev nodes](https://bliptext.com/articles/chebyshev-nodes) are used to construct the even order modified Chebyshev polynomials.\nwhere\nFor example, the 4th order Chebyshev polynomial from the [example above](https://bliptext.com/articles/chebyshev-polynomials) is, which by inspection contains no roots of zero.\nCreating the polynomial from the even order modified Chebyshev nodes creates a 4th order even order modified Chebyshev polynomial of, which by inspection contains two roots at zero, and may be used in applications requiring roots at zero.\n\n# Sources","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T18:34:48.673Z","updated_at":"2025-01-25T18:34:48.673Z","search_vector":"'chebyshev':1 'polynomi':2","search_content":" \n \n\n# \nThe **Chebyshev [polynomials](https://bliptext.com/articles/polynomial)** are two sequences of [polynomials](https://bliptext.com/articles/polynomial) related to the [cosine and sine functions](https://bliptext.com/articles/trigonometric-functions), notated as T_n(x) and U_n(x).\nThey can be defined in several equivalent ways, one of which starts with [trigonometric functions](https://bliptext.com/articles/trigonometric-functions):\nThe **Chebyshev polynomials of the first kind** T_n are defined by:\nSimilarly, the **Chebyshev polynomials of the second kind** U_n are defined by:\nThat these expressions define polynomials in \\cos\\theta may not be obvious at first sight but follows by rewriting and using [de Moivre's formula](https://bliptext.com/articles/de-moivre-s-formula) or by using the [angle sum formulas](https://bliptext.com/articles/list-of-trigonometric-identities) for \\cos and \\sin repeatedly.\nFor example, the [double angle formulas](https://bliptext.com/articles/list-of-trigonometric-identities), which follow directly from the angle sum formulas, may be used to obtain and, which are respectively a polynomial in \\cos\\theta and a polynomial in \\cos\\theta multiplied by \\sin\\theta.\nHence and U_1(x) = 2x.\nAn important and convenient property of the\nTn(x)\nis that they are orthogonal with respect to the following [inner product](https://bliptext.com/articles/inner-product):\nand\nUn(x)\nare orthogonal with respect to another, analogous inner product, given below.\nThe Chebyshev polynomials\nTn\nare polynomials with the largest possible leading coefficient whose [absolute value](https://bliptext.com/articles/absolute-value) on the [interval](https://bliptext.com/articles/interval-mathematics) is bounded by 1.\nThey are also the \"extremal\" polynomials for many other properties.\nIn 1952, [Cornelius Lanczos](https://bliptext.com/articles/cornelius-lanczos) showed that the Chebyshev polynomials are important in [approximation theory](https://bliptext.com/articles/approximation-theory) for the solution of linear systems; the [roots](https://bliptext.com/articles/root-of-a-polynomial) of\nTn(x)\n, which are also called [Chebyshev nodes](https://bliptext.com/articles/chebyshev-nodes), are used as matching points for optimizing [polynomial interpolation](https://bliptext.com/articles/polynomial-interpolation).\nThe resulting interpolation polynomial minimizes the problem of [Runge's phenomenon](https://bliptext.com/articles/runge-s-phenomenon) and provides an approximation that is close to the best polynomial approximation to a [continuous function](https://bliptext.com/articles/continuous-function) under the [maximum norm](https://bliptext.com/articles/maximum-norm), also called the \"[minimax](https://bliptext.com/articles/minimax)\" criterion.\nThis approximation leads directly to the method of [Clenshaw–Curtis quadrature](https://bliptext.com/articles/clenshaw-curtis-quadrature).\nThese polynomials were named after [Pafnuty Chebyshev](https://bliptext.com/articles/pafnuty-chebyshev).\nThe letter T is used because of the alternative [transliterations](https://bliptext.com/articles/transliteration) of the name Chebyshev as Tchebycheff, Tchebyshev (French) or Tschebyschow (German).\n# [Chebyshev polynomials | upload.wikimedia.org/wikipedia/commons/4/4d/Plot///of///the///Chebyshev///polynomial///of///the///first///kind///T///n%28x%29///with///n%3D5///in///the///complex///plane///from///-2-2i///to///2%2B2i// /with///colors///created///with///Mathematica///13.1///function///ComplexPlot3D.svg]\n\n# Definitions\n\n## Recurrence definition\nThe **Chebyshev polynomials of the first kind** are obtained from the [recurrence relation](https://bliptext.com/articles/recurrence-relation):\nThe recurrence also allows to represent them explicitly as the determinant of a [tridiagonal matrix](https://bliptext.com/articles/tridiagonal-matrix) of size k \\times k:\nThe [ordinary generating function](https://bliptext.com/articles/generating-function) for Tn is:\nThere are several other [generating functions](https://bliptext.com/articles/generating-function) for the Chebyshev polynomials; the [exponential generating function](https://bliptext.com/articles/exponential-generating-function) is:\nThe generating function relevant for 2-dimensional [potential theory](https://bliptext.com/articles/potential-theory) and [multipole expansion](https://bliptext.com/articles/cylindrical-multipole-moments) is:\nThe **Chebyshev polynomials of the second kind** are defined by the recurrence relation:\nNotice that the two sets of recurrence relations are identical, except for T_1(x) = x vs. The ordinary generating function for Un is:\nand the exponential generating function is:\n# [Plot of the first five Tn Chebyshev polynomials (first kind) | upload.wikimedia.org/wikipedia/commons/5/5c/Chebyshev///Polynomials///of///the///First///Kind.svg]\n# [Plot of the first five Un Chebyshev polynomials (second kind) | upload.wikimedia.org/wikipedia/commons/5/53/Chebyshev///Polynomials///of///the///Second///Kind.svg]\n\n## Trigonometric definition\nAs described in the introduction, the Chebyshev polynomials of the first kind can be defined as the unique polynomials satisfying:\nor, in other words, as the unique polynomials satisfying:\nfor\nn = 0, 1, 2, 3, …\n.\nThe polynomials of the second kind satisfy:\nor\nwhich is structurally quite similar to the [Dirichlet kernel](https://bliptext.com/articles/dirichlet-kernel)\nDn(x)\n(The Dirichlet kernel, in fact, coincides with what is now known as the [Chebyshev polynomial of the fourth kind](https://bliptext.com/articles/).)\nAn equivalent way to state this is via exponentiation of a [complex number](https://bliptext.com/articles/complex-number): given a [complex number](https://bliptext.com/articles/complex-number)\nz = a + bi\nwith absolute value of one:\nChebyshev polynomials can be defined in this form when studying [trigonometric polynomials](https://bliptext.com/articles/trigonometric-polynomials).\nThat\ncos nx\nis an nth-[degree](https://bliptext.com/articles/degree-of-a-polynomial) polynomial in\ncos x\ncan be seen by observing that\ncos nx\nis the [real part](https://bliptext.com/articles/complex-number) of one side of [de Moivre's formula](https://bliptext.com/articles/de-moivre-s-formula):\nThe real part of the other side is a polynomial in\ncos x\nand\nsin x\n, in which all powers of\nsin x\nare [even](https://bliptext.com/articles/parity-mathematics) and thus replaceable through the identity\ncos2 x + sin2 x = 1\n. By the same reasoning,\nsin nx\nis the [imaginary part](https://bliptext.com/articles/complex-number) of the polynomial, in which all powers of\nsin x\nare [odd](https://bliptext.com/articles/parity-mathematics) and thus, if one factor of\nsin x\nis factored out, the remaining factors can be replaced to create a\n(n−1)\nst-degree polynomial in\ncos x\n.\n\n## Commuting polynomials definition\nChebyshev polynomials can also be characterized by the following theorem:\nIf F_n(x) is a family of monic polynomials with coefficients in a field of characteristic 0 such that and for all\nm and n, then, up to a simple change of variables, either for all n or for all n.\n\n## Pell equation definition\nThe Chebyshev polynomials can also be defined as the solutions to the [Pell equation](https://bliptext.com/articles/pell-equation):\nin a [ring](https://bliptext.com/articles/ring-mathematics)\nR[x]\n. Thus, they can be generated by the standard technique for Pell equations of taking powers of a fundamental solution:\n\n# Relations between the two kinds of Chebyshev polynomials\nThe Chebyshev polynomials of the first and second kinds correspond to a complementary pair of [Lucas sequences](https://bliptext.com/articles/lucas-sequence)\nṼn(P, Q)\nand\nŨn(P, Q)\nwith parameters\nP = 2x\nand\nQ = 1\nIt follows that they also satisfy a pair of mutual recurrence equations:\nThe second of these may be rearranged using the [recurrence definition](https://bliptext.com/articles/) for the Chebyshev polynomials of the second kind to give:\nUsing this formula iteratively gives the sum formula:\nwhile replacing U_n(x) and U_{n-2}(x) using the [derivative formula](https://bliptext.com/articles/) for T_n(x) gives the recurrence relationship for the derivative of T_n:\nThis relationship is used in the [Chebyshev spectral method](https://bliptext.com/articles/chebyshev-spectral-method) of solving differential equations.\n[Turán's inequalities](https://bliptext.com/articles/tur-n-s-inequalities) for the Chebyshev polynomials are:\nThe [integral](https://bliptext.com/articles/integral) relations are\nwhere integrals are considered as principal value.\n\n# Explicit expressions\nDifferent approaches to defining Chebyshev polynomials lead to different explicit expressions.\nThe trigonometric definition gives an explicit formula as follows:\nFrom this trigonometric form, the recurrence definition can be recovered by computing directly that the bases cases hold:\nand\nand that the [product-to-sum identity](https://bliptext.com/articles/list-of-trigonometric-identities) holds:\nUsing the complex number exponentiation definition of the Chebyshev polynomial, one can derive the following expression:\nThe two are equivalent because.\nAn explicit form of the Chebyshev polynomial in terms of monomials\nxk\nfollows from [de Moivre's formula](https://bliptext.com/articles/de-moivre-s-formula):\nwhere\nRe\ndenotes the [real part](https://bliptext.com/articles/complex-number) of a complex number.\nExpanding the formula, one gets:\nThe real part of the expression is obtained from summands corresponding to even indices.\nNoting and, one gets the explicit formula:\nwhich in turn means that:\nThis can be written as a\n2F1\n[hypergeometric function](https://bliptext.com/articles/hypergeometric-function):\nwith inverse:\nwhere the prime at the summation symbol indicates that the contribution of\nneeds to be halved if it appears.\nA related expression for\nTn\nas a sum of monomials with binomial coefficients and powers of two is\nSimilarly,\nUn\ncan be expressed in terms of hypergeometric functions:\n\n# Properties\n\n## Symmetry\nThat is, Chebyshev polynomials of even order have [even symmetry](https://bliptext.com/articles/even-and-odd-functions) and therefore contain only even powers of x. Chebyshev polynomials of odd order have [odd symmetry](https://bliptext.com/articles/even-and-odd-functions) and therefore contain only odd powers of x.\n\n## Roots and extrema\nA Chebyshev polynomial of either kind with degree n has n different [simple roots](https://bliptext.com/articles/simple-root), called **Chebyshev roots**, in the interval.\nThe roots of the Chebyshev polynomial of the first kind are sometimes called [Chebyshev nodes](https://bliptext.com/articles/chebyshev-nodes) because they are used as nodes in polynomial interpolation.\nUsing the trigonometric definition and the fact that:\none can show that the roots of Tn are:\nSimilarly, the roots of Un are:\nThe [extrema](https://bliptext.com/articles/maxima-and-minima) of Tn on the interval\n−1 ≤ x ≤ 1\nare located at:\nOne unique property of the Chebyshev polynomials of the first kind is that on the interval\n−1 ≤ x ≤ 1\nall of the [extrema](https://bliptext.com/articles/maxima-and-minima) have values that are either −1 or 1.\nThus these polynomials have only two finite [critical values](https://bliptext.com/articles/critical-value-critical-point), the defining property of [Shabat polynomials](https://bliptext.com/articles/shabat-polynomial).\nBoth the first and second kinds of Chebyshev polynomial have extrema at the endpoints, given by:\nThe [extrema](https://bliptext.com/articles/maxima-and-minima) of T_n(x) on the interval where n>0 are located at n+1 values of x. They are \\pm 1, or where d > 2, d \\;|\\; 2n, 0 i + j\nwith the same Chebyshev nodes\nxk\n, there are similar sums:\nand without the weight function:\nFor any integer\nN > i + j\n, based on the N zeros of\nUN (x)\none can get the sum:\nand again without the weight function:\n\n## Minimal \n\n∞\n\n-norm\nFor any given\nn ≥ 1\n, among the polynomials of degree n with leading coefficient 1 ([monic](https://bliptext.com/articles/monic-polynomial) polynomials):\nis the one of which the maximal absolute value on the interval is minimal.\nThis maximal absolute value is:\nand\nreaches this maximum exactly\nn + 1\ntimes at:\n\n### Remark\nBy the [equioscillation theorem](https://bliptext.com/articles/equioscillation-theorem), among all the polynomials of degree\n≤ n\n, the polynomial f minimizes\n f ∞\non [if and only if](https://bliptext.com/articles/if-and-only-if) there are\nn + 2\npoints\n−1 ≤ x0 < x1 < ⋯ < xn + 1 ≤ 1\nsuch that\n= f ∞\n.\nOf course, the null polynomial on the interval can be approximated by itself and minimizes the\n∞\n-norm.\nAbove, however,\nreaches its maximum only\nn + 1\ntimes because we are searching for the best polynomial of degree\nn ≥ 1\n(therefore the theorem evoked previously cannot be used).\n\n## Chebyshev polynomials as special cases of more general polynomial families\nThe Chebyshev polynomials are a special case of the ultraspherical or [Gegenbauer polynomials](https://bliptext.com/articles/gegenbauer-polynomials), which themselves are a special case of the [Jacobi polynomials](https://bliptext.com/articles/jacobi-polynomials) :\nChebyshev polynomials are also a special case of [Dickson polynomials](https://bliptext.com/articles/dickson-polynomial):\nIn particular, when, they are related by and.\n\n## Other properties\nThe curves given by\n, or equivalently, by the parametric equations\n,\n, are a special case of [Lissajous curves](https://bliptext.com/articles/lissajous-curve) with frequency ratio equal to n.\nSimilar to the formula:\nwe have the analogous formula:\nFor\nx ≠ 0\nand:\nwhich follows from the fact that this holds by definition for\nx = eiθ\n.\nThere are relations between Legendre polynomials and Chebyshev polynomials\nThese identities can be proven using generating functions and discrete convolution\n\n# Examples\n\n## First kind\n[[File:Chebyshev Polynomials of the 1st Kind (n=0-5, x=(-1,1)).svg|thumb|300px|The first few Chebyshev polynomials of the first kind in the domain\n−1 < x < 1\nThe flat\nT0\n,\nT1\n,\nT2\n,\nT3\n,\nT4\nand\nT5\n.]]\nThe first few Chebyshev polynomials of the first kind are\n\n## Second kind\n[[File:Chebyshev Polynomials of the 2nd Kind (n=0-5, x=(-1,1)).svg|thumb|300px|The first few Chebyshev polynomials of the second kind in the domain\n−1 < x < 1\nU0\n,\nU1\n,\nU2\n,\nU3\n,\nU4\nand\nU5\n. Although not visible in the image,\nUn(1) = n + 1\nand\nUn(−1) = (n + 1)(−1)n\n.]]\nThe first few Chebyshev polynomials of the second kind are\n\n# As a basis set\n[[Image:ChebyshevExpansion.png|thumb|right|240px|The non-smooth function (top)\ny = −x3H(−x)\n, where H is the [Heaviside step function](https://bliptext.com/articles/heaviside-step-function), and (bottom) the 5th partial sum of its Chebyshev expansion.\nThe 7th sum is indistinguishable from the original function at the resolution of the graph.]]\nIn the appropriate [Sobolev space](https://bliptext.com/articles/sobolev-space), the set of Chebyshev polynomials form an [orthonormal basis](https://bliptext.com/articles/hilbert-space), so that a function in the same space can, on\n−1 ≤ x ≤ 1\n, be expressed via the expansion:\nFurthermore, as mentioned previously, the Chebyshev polynomials form an [orthogonal](https://bliptext.com/articles/orthogonal) basis which (among other things) implies that the coefficients\nan\ncan be determined easily through the application of an [inner product](https://bliptext.com/articles/inner-product).\nThis sum is called a **Chebyshev series** or a **Chebyshev expansion**.\nSince a Chebyshev series is related to a [Fourier cosine series](https://bliptext.com/articles/fourier-cosine-series) through a change of variables, all of the theorems, identities, etc. that apply to [Fourier series](https://bliptext.com/articles/fourier-series) have a Chebyshev counterpart.\nThese attributes include:\nf(x)\nif the function is [piecewise](https://bliptext.com/articles/piecewise) [smooth](https://bliptext.com/articles/smooth-function) and [continuous](https://bliptext.com/articles/continuous-function).\nThe smoothness requirement can be relaxed in most cases – as long as there are a finite number of discontinuities in\nf(x)\nand its derivatives.\nThe abundance of the theorems and identities inherited from [Fourier series](https://bliptext.com/articles/fourier-series) make the Chebyshev polynomials important tools in [numeric analysis](https://bliptext.com/articles/numeric-analysis); for example they are the most popular general purpose basis functions used in the [spectral method](https://bliptext.com/articles/spectral-method), often in favor of trigonometric series due to generally faster convergence for continuous functions ([Gibbs' phenomenon](https://bliptext.com/articles/gibbs-phenomenon) is still a problem).\n\n## Example 1\nConsider the Chebyshev expansion of\nlog(1 + x)\n. One can express:\nOne can find the coefficients\nan\neither through the application of an inner product or by the discrete orthogonality condition.\nFor the inner product:\nwhich gives:\nAlternatively, when the inner product of the function being approximated cannot be evaluated, the discrete orthogonality condition gives an often useful result for approximate coefficients:\nwhere δij is the [Kronecker delta](https://bliptext.com/articles/kronecker-delta) function and the xk are the N Gauss–Chebyshev zeros of\nTN (x)\nFor any N, these approximate coefficients provide an exact approximation to the function at xk with a controlled error between those points.\nThe exact coefficients are obtained with\nN = ∞\n, thus representing the function exactly at all points in.\nThe rate of convergence depends on the function and its smoothness.\nThis allows us to compute the approximate coefficients an very efficiently through the [discrete cosine transform](https://bliptext.com/articles/discrete-cosine-transform):\n\n## Example 2\nTo provide another example:\n\n## Partial sums\nThe partial sums of:\nare very useful in the [approximation](https://bliptext.com/articles/approximation-theory) of various functions and in the solution of [differential equations](https://bliptext.com/articles/differential-equation) (see [spectral method](https://bliptext.com/articles/spectral-method)).\nTwo common methods for determining the coefficients an are through the use of the [inner product](https://bliptext.com/articles/inner-product) as in [Galerkin's method](https://bliptext.com/articles/galerkin-s-method) and through the use of [collocation](https://bliptext.com/articles/collocation-method) which is related to [interpolation](https://bliptext.com/articles/interpolation).\nAs an interpolant, the N coefficients of the\n(N − 1)\nst partial sum are usually obtained on the Chebyshev–Gauss–Lobatto points (or Lobatto grid), which results in minimum error and avoids [Runge's phenomenon](https://bliptext.com/articles/runge-s-phenomenon) associated with a uniform grid.\nThis collection of points corresponds to the extrema of the highest order polynomial in the sum, plus the endpoints and is given by:\n\n## Polynomial in Chebyshev form\nAn arbitrary polynomial of degree N can be written in terms of the Chebyshev polynomials of the first kind.\nSuch a polynomial\np(x)\nis of the form:\nPolynomials in Chebyshev form can be evaluated using the [Clenshaw algorithm](https://bliptext.com/articles/clenshaw-algorithm).\n\n# Families of polynomials related to Chebyshev polynomials\nPolynomials denoted C_n(x) and S_n(x) closely related to Chebyshev polynomials are sometimes used.\nThey are defined by:\nand satisfy:\nA. F. Horadam called the polynomials C_n(x) **Vieta–Lucas polynomials** and denoted them v_n(x).\nHe called the polynomials S_n(x) **Vieta–Fibonacci polynomials** and denoted them V_n(x).\nLists of both sets of polynomials are given in [Viète's](https://bliptext.com/articles/fran-ois-vi-te) Opera Mathematica, Chapter IX, Theorems VI and VII.\nThe Vieta–Lucas and Vieta–Fibonacci polynomials of real argument are, up to a power of i and a shift of index in the case of the latter, equal to [Lucas and Fibonacci polynomials](https://bliptext.com/articles/fibonacci-polynomials)\nLn\nand\nFn\nof imaginary argument.\n**Shifted Chebyshev polynomials** of the first and second kinds are related to the Chebyshev polynomials by:\nWhen the argument of the Chebyshev polynomial satisfies\n2x − 1 ∈\nthe argument of the shifted Chebyshev polynomial satisfies\nx ∈\n. Similarly, one can define shifted polynomials for generic intervals.\nAround 1990 the terms \"third-kind\" and \"fourth-kind\" came into use in connection with Chebyshev polynomials, although the polynomials denoted by these terms had an earlier development under the name **airfoil polynomials**.\nAccording to J. C. Mason and G. H. Elliott, the terminology \"third-kind\" and \"fourth-kind\" is due to [Walter Gautschi](https://bliptext.com/articles/walter-gautschi), \"in consultation with colleagues in the field of orthogonal polynomials.\"\nThe **Chebyshev polynomials of the third kind** are defined as:\nand the **Chebyshev polynomials of the fourth kind** are defined as:\nwhere.\nIn the airfoil literature V_n(x) and W_n(x) are denoted t_n(x) and u_n(x).\nThe polynomial families T_n(x), U_n(x), V_n(x), and W_n(x) are orthogonal with respect to the weights:\nand are proportional to Jacobi polynomials with:\nAll four families satisfy the recurrence with p_0(x) = 1, where p_n = T_n, U_n, V_n, or W_n, but they differ according to whether p_1(x) equals x, 2x, 2x-1, or 2x+1.\n\n## Even order modified Chebyshev polynomials\nSome applications rely on Chebyshev polynomials but may be unable to accommodate the lack of a root at zero, which rules out the use of standard Chebyshev polynomials for these kinds of applications.\nEven order Chebyshev filter designs using equally terminated passive networks are an example of this.\nHowever, even order Chebyshev polynomials may be modified to move the lowest roots down to zero while still maintaining the desirable Chebyshev equi-ripple effect.\nSuch modified polynomials contain two roots at zero, and may be referred to as even order modified Chebyshev polynomials.\nEven order modified Chebyshev polynomials may be created from the [Chebyshev nodes](https://bliptext.com/articles/chebyshev-nodes) in the same manner as standard Chebyshev polynomials.\nwhere\nIn the case of even order modified Chebyshev polynomials, the [even order modified Chebyshev nodes](https://bliptext.com/articles/chebyshev-nodes) are used to construct the even order modified Chebyshev polynomials.\nwhere\nFor example, the 4th order Chebyshev polynomial from the [example above](https://bliptext.com/articles/chebyshev-polynomials) is, which by inspection contains no roots of zero.\nCreating the polynomial from the even order modified Chebyshev nodes creates a 4th order even order modified Chebyshev polynomial of, which by inspection contains two roots at zero, and may be used in applications requiring roots at zero.\n\n# Sources","revisionCount":0},"hasMore":false,"nextCursor":null}

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