Buy Chebyshev Polynomials: From Approximation Theory to Algebra and Number Theory (Pure and Applied Mathematics: A Wiley Theodore J. Rivlin ( Author). Rivlin, an introduction to the approximation of functions blaisdell, qa A note on chebyshev polynomials, cyclotomic polynomials and. Wiscombe. (Rivlin  gives numer- ous examples.) Their significance can be immediately appreciated by the fact that the function cosnθ is a Chebyshev polynomial function.
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In mathematics the Chebyshev polynomialsnamed after Pafnuty Chebyshev are a sequence of orthogonal polynomials which are related to de Moivre’s formula and which can be defined recursively. One usually distinguishes between Chebyshev polynomials of the first kind which are denoted T n and Chebyshev polynomials of the second kind which are denoted U n.
The letter T is used because of the alternative transliterations of the name Chebyshev as TchebycheffTchebyshev French or Tschebyschow German. The Chebyshev polynomials T n or U n are polynomials of degree n and the sequence of Chebyshev polynomials of either kind composes a polynomial sequence.
They are also the extremal polynomials for many other properties. Chebyshev polynomials are important in approximation theory because the roots of the Chebyshev polynomials of the first kind, which are also called Chebyshev nodesare used as nodes in polynomial interpolation.
The resulting interpolation polynomial minimizes the problem of Runge’s phenomenon and provides an approximation that is close to the polynomial of best approximation to a continuous function under the maximum norm. This approximation leads directly to the method of Clenshaw—Curtis quadrature. In the study of differential equations they arise as the solution to the Chebyshev differential equations.
These equations are special cases of the Sturm—Liouville differential equation. The Chebyshev polynomials of the first kind are defined by the recurrence relation. The ordinary generating function for T n is. The generating function relevant for 2-dimensional potential theory and multipole expansion is. The ordinary generating function for U n is. T n x is functionally conjugate to nxcodified in the nesting property below. Further compare to the spread polynomialsin the section below. That cos nx is an n th-degree polynomial in cos x can be seen by observing that cos nx is the real part of one side of de Moivre’s formula.
By the same reasoning, sin nx is the imaginary part of the polynomial, in which all powers of sin x are odd and thus, if one is factored out, the remaining can be replaced to create a n-1 th-degree polynomial in cos x. The identity is quite useful in conjunction with the recursive generating formula, inasmuch as it enables one to calculate the cosine of any integral multiple of an angle solely in terms of the cosine of the base angle.
Two immediate corollaries are the composition identity or nesting property specifying a semigroup. The Chebyshev polynomials can also be defined as the solutions to the Pell equation. When working with Chebyshev polynomials quite often products of two of them occur.
These products can be reduced to combinations of Chebyshev polynomials with lower or higher degree and concluding statements about the product are easier to make. It shall be assumed that in the following the index m is greater than or equal to the index n and n is not negative.
For Chebyshev polynomials of the first kind the product expands to. Three more useful formulas for evaluating Chebyshev polynomials can be concluded from this product expansion:. The Chebyshev polynomials of the first and second kinds are also connected by the following relations:. The recurrence relationship of the derivative of Chebyshev polynomials can be derived from these relations:.
This relationship is used in the Chebyshev spectral method of solving differential equations. Different approaches to defining Chebyshev polynomials lead to different explicit expressions such as:.
That is, Chebyshev polynomials of even order have even symmetry and contain only even powers of x. Chebyshev polynomials of odd order have odd symmetry and contain only odd powers of x.
The roots of the Chebyshev polynomial of the first kind are sometimes called Chebyshev nodes because they are used as nodes in polynomial interpolation. Using the trigonometric definition and the fact that. Similarly, the roots of U n are. Thus these polynomials have only two finite critical valuesthe defining property of Shabat polynomials. Both the first and second kinds of Chebyshev polynomial have extrema at the endpoints, given by:.
The derivatives of the polynomials can be less than straightforward. By differentiating the polynomials in their trigonometric forms, it’s easy to show that:.
It can be shown that:. The second derivative of the Chebyshev polynomial of the first kind is. Since the function is a polynomial, all of the derivatives must exist for all real numbers, so the taking to limit on the expression above should yield the desired value:. Since the limit as a whole must exist, the limit of the numerator and denominator must independently exist, and.
The denominator still limits to zero, which implies that the numerator must be limiting to zero, i. Concerning integration, the first derivative of the T n implies that. Both T n and U n form a sequence of orthogonal polynomials. The polynomials of the first kind T n are orthogonal with respect to the weight. Similarly, the polynomials of the second kind U n are orthogonal with respect to the weight.
The T n also satisfy a discrete orthogonality condition:. For the polynomials of the second kind and with the same Chebyshev nodes x k there are similar sums:. Based on the N zeros of the Chebyshev polynomial of the second kind U N x:. Because at extreme points of T n we have.
From the intermediate value theoremf n x has at least n roots. The Chebyshev polynomials are a special case of the ultraspherical or Gegenbauer polynomialschebysyev themselves are polymomials special case of the Jacobi polynomials:.
For every nonnegative integer nT n x and U n x are both polynomials of degree n. They are even or odd functions of x as n is even or odd, so when written as polynomials of xit only has even or odd degree terms respectively. T n are a special case of Lissajous curves with frequency ratio equal to n. The polynomials of the second polynomiaos satisfy the similar relation. Then C n x and C m x are commuting polynomials:.
They have the power series expansion. Furthermore, as mentioned previously, the Chebyshev polynomials form an orthogonal basis which among other things implies that the coefficients a n can be determined easily through the application of an inner product. This sum is called a Chebyshev series or a Chebyshev expansion.
Since a Chebyshev series is related to a Fourier cosine series through a change of variables, all of the theorems, identities, etc. The abundance of the theorems and identities inherited from Fourier series make the Chebyshev polynomials important tools in numeric analysis ; for example they are the most popular general purpose basis functions used in the spectral method often in favor of trigonometric series due to generally faster convergence for continuous functions Divlin phenomenon is still a problem.
The Chebyshev Polynomials – Theodore J. Rivlin – Google Books
One can find the coefficients a n either through the application of an inner product or by the discrete orthogonality condition. For the inner product. Alternatively, when you cannot evaluate the inner product of the function you are trying to approximate, the discrete orthogonality condition gives an often useful result for approximate coefficients.
For any Nthese approximate coefficients provide an exact approximation to polyonmials function at x k with a controlled error between those points. The rate of convergence depends on the function and its smoothness. This allows us to compute the approximate coefficients a n very efficiently through the discrete cosine transform. Two common methods for determining the coefficients a n are through the use of the inner product as in Galerkin’s method and through the use of collocation which is related to interpolation.
This collection of points corresponds to the extrema of the highest order polynomial in the sum, plus the endpoints and is given by:. An arbitrary polynomial of degree N can be written in terms of the Chebyshev polynomials eivlin the first kind. Polynomials in Chebyshev form can be evaluated using the Clenshaw algorithm.
Similarly, one can define shifted polynomials for generic intervals [ ab ]. The spread polynomials are a rescaling of the shifted Chebyshev polynomials of the first kind so that the range is also [0,1]. From Wikipedia, the free encyclopedia.
Not to be confused with discrete Chebyshev polynomials. Since the function is a polynomial, all of the derivatives must exist for all real numbers, so the taking to limit on the expression above should yield the desired value: Pure and Applied Mathematics.
Chapter 2, “Extremal Properties”, pp.