18
Chebyshev Polynomials
Abstract
The topics of this chapter are Chebyshev polynomials of 1st, 2nd, 3rd, and 4th
kind. Arbitrarily shifted polynomials will be discussed as well as extensions
into the complex domain. The polynomials can be generalized to functions
with complex degree. The evaluation could be based on recurrence relations,
on hypergeometric functions or on trigonometric series with generalizations to
the complex domain via conformal mapping. Code for evaluating the Chebyshev
polynomials, respectively, functions and their nodes can be downloaded. Additional methods of orthogonal polynomials are supported.
The topics of this chapter are Chebyshev polynomials, named after Pafnuty Chebyshev, a Russian nineteenth century mathematician. In literature there are alternative
transliterations of his name used, e.g., Tschebyscheff or Tchebyshev.
Chebyshev polynomials are orthogonal polynomials and play an important rôle,
e.g., in polynomial interpolation and approximation theory, or in solving integral
equations, filter techniques and constructing wavelets.
Function Overview
The SPECFUNPHYS class chebypoly returns the polynomial coefficients of the
Chebyshev polynomials and shifted Chebyshev polynomials of 1st, 2nd, 3rd, and
4th kind. The SPECFUNPHYS class chebyx supports the computation of the
Chebyshev polynomials, the shifted Chebyshev polynomials and arbitrarily shifted
polynomials and functions of 1st, 2nd, 3rd, and 4th kind based on different methods.
The Chebyshev polynomials can be extended to functions (non-integer degree) and
the complex domain. The SPECFUNPHYS function chebyzero returns the nodes
of the Chebyshev polynomials.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
W. Schweizer, Special Functions in Physics with MATLAB,
https://doi.org/10.1007/978-3-030-64232-7_18
215
Chebyshev Polynomials
Abstract
The topics of this chapter are Chebyshev polynomials of 1st, 2nd, 3rd, and 4th
kind. Arbitrarily shifted polynomials will be discussed as well as extensions
into the complex domain. The polynomials can be generalized to functions
with complex degree. The evaluation could be based on recurrence relations,
on hypergeometric functions or on trigonometric series with generalizations to
the complex domain via conformal mapping. Code for evaluating the Chebyshev
polynomials, respectively, functions and their nodes can be downloaded. Additional methods of orthogonal polynomials are supported.
The topics of this chapter are Chebyshev polynomials, named after Pafnuty Chebyshev, a Russian nineteenth century mathematician. In literature there are alternative
transliterations of his name used, e.g., Tschebyscheff or Tchebyshev.
Chebyshev polynomials are orthogonal polynomials and play an important rôle,
e.g., in polynomial interpolation and approximation theory, or in solving integral
equations, filter techniques and constructing wavelets.
Function Overview
The SPECFUNPHYS class chebypoly returns the polynomial coefficients of the
Chebyshev polynomials and shifted Chebyshev polynomials of 1st, 2nd, 3rd, and
4th kind. The SPECFUNPHYS class chebyx supports the computation of the
Chebyshev polynomials, the shifted Chebyshev polynomials and arbitrarily shifted
polynomials and functions of 1st, 2nd, 3rd, and 4th kind based on different methods.
The Chebyshev polynomials can be extended to functions (non-integer degree) and
the complex domain. The SPECFUNPHYS function chebyzero returns the nodes
of the Chebyshev polynomials.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
W. Schweizer, Special Functions in Physics with MATLAB,
https://doi.org/10.1007/978-3-030-64232-7_18
215