Chebyshev polynomials
Chebyshev polynomials, named after Pafnuti Lvovitch Chebyshev in the literature as Chebyshev, Chebyshev, Chebyshev, Chebyshev or Chebyshev called, are recursive polynomials in mathematics. A distinction is made between Chebyshev polynomials of the first kind Chebyshev polynomials of the second kind.
Chebyshev polynomials of the first kind are solution of the Chebyshev differential equation
And Chebyshev polynomials of the second kind are 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.
For each integer violates either of these series after a finite number of terms from, for even and odd, and one obtains polynomials as a solution. With the normalization these are called Chebyshev polynomials. The first seven polynomials of this type are:
You can in a general way from the recursive relation
Be calculated. With the help of the trigonometric functions and the hyperbolic functions the Chebyshev polynomials are represented as
Or
The zeros of the Chebyshev polynomial are given by
Chebyshev polynomials are orthogonal with respect to the closed interval of the weighted scalar product
One can therefore this via the Gram -Schmidt orthogonalization ( with normalization) derive.
Applications
In the filter technique, the Chebyshev polynomials are used in the Chebyshev filters. In polynomial interpolation, these polynomials are characterized by a very low and uniform error profile. These are to be used as interpolating the appropriate shifted zeros of the Chebyshev polynomial matching degree. Because of their minimality they also form the basis for the Chebyshev iteration and error bounds for Krylov subspace methods for linear systems of equations.
Chebyshev polynomials of the second kind
The Chebyshev polynomials of the second kind are defined by a recursive formation rule:
The generating function for is:
The first eight polynomials of this type are:
Chebyshev polynomials are orthogonal with respect to the closed interval of the weighted scalar product
History
First published his studies on the Chebyshev Chebyshev polynomials 1859 and 1881 in the following attachments:
- Sur les questions qui se de minima rattachent a la représentation approximative des fonctions, 1859, Oeuvres, Volume I, page 273-378
- Sur les fonctions qui peu de s'écartent zéro pour certaines valeurs de la variable 1881, Oeuvres Volume II, page 335-356