Chebyshev polynomial

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Chebyshev polynomials of the first type and the second type are sequences of orthogonal polynomials that have important applications in polynomial interpolation , in filter technology, and in other areas of mathematics. They are named after Pafnuti Lwowitsch Tschebyschow , whose name is also transcribed in the literature as Chebyshev, Chebyshev, Chebyshev, Chebyshev, Chebyshev or Chebychev .

Chebyshev polynomials of the first kind are the solution of the Chebyshev differential equation

and Chebyshev polynomials of the second kind are the solution of

Both differential equations are special cases of the Sturm-Liouville differential equation .

Chebyshev polynomials of the first kind

The functions

and

form a fundamental system for the Chebyshev differential equation.

Chebyshev polynomials of the first kind of order 0 to 5.

For integer numbers, one of these series breaks off after a finite number of terms, for even and for odd , and you get polynomials as a solution. With the normalization these are referred to as Chebyshev polynomials . The first nine polynomials of this type are:

You can get out of the recursive context in a general way

be calculated. With the help of the trigonometric functions or the hyperbolic functions , the Chebyshev polynomials can be represented as

or

and also

.

The zeros of the Chebyshev polynomial are given by

Chebyshev polynomials are orthogonal in the closed interval with respect to the weighted scalar product

This can therefore also be derived from the Gram-Schmidt orthogonalization method (with normalization).

Applications

In filter technology , the Chebyshev polynomials are used in Chebyshev filters . In the case of polynomial interpolation , these polynomials are characterized by a very favorable, uniform error curve. For this purpose, the suitably shifted zeros of the Chebyshev polynomial of the appropriate degree are to be used as interpolation points. Because of their minimality, they also form the basis for the Chebyshev iteration and for error bounds in Krylov subspace methods for systems of linear equations .

Chebyshev polynomials of the second kind

Chebyshev polynomials of the second kind of order 0 to 5.

The Chebyshev polynomials of the second kind are also defined using a recursive formation rule:

notably with the same recursion relationship as the . And this recursion relationship also applies

 

also for .

The generating function for is:

The first eight polynomials of this type are:

With the help of the trigonometric functions , the Chebyshev polynomials of the second type can initially only be represented as

but because of the constant liftability at these points for everyone . This formula is structurally similar to the Dirichlet core :

If you add hyperbolic functions, then it is for

Chebyshev polynomials are orthogonal to the weighted scalar product in the closed interval

history

For the first time, Chebyshev published his research on the Chebyshev polynomials in 1859 and 1881 in the following articles:

  • Sur les questions de minima qui se rattachent a la représentation approximative des fonctions , 1859, Oeuvres Volume I, pages 273–378
  • Sur les fonctions qui s'écartent peu de zéro pour certaines valeurs de la variable , 1881, Oeuvres Volume II, pages 335–356

Clenshaw algorithm

In numerical mathematics, linear combinations of Chebyshev polynomials are evaluated using the Clenshaw algorithm.

literature

Web links

Individual evidence

  1. Leçons sur l'approximation des fonctions d'une variable réellehttp: //vorlage_digitalisat.test/1%3D~GB%3D~IA%3Dleonssurlappro00lavauoft~MDZ%3D%0A~SZ%3D~doppelseiten%3D~LT%3D%27%27Le%C3%A7ons%20sur%20l% 27approximation% 20des% 20fonctions% 20d% 27une% 20variable% 20r% C3% A9elle% 27% 27 ~ PUR% 3D Paris, Gauthier-Villars, 1919, 1952. Page 64
  2. ^ Elliot Ward Cheney: Introduction to Approximation Theory , McGraw-Hill Book Company, 1966, Library of Congress Catalog Card Number 65-25916, ISBN 007-010757-2 , p. 225