Properties of Chebyshev polynomials

TL;DR

This paper reviews key properties of Chebyshev polynomials, emphasizing their applications in solving differential equations and their superior accuracy in polynomial approximation compared to Fourier series.

Contribution

It provides a comprehensive overview of Chebyshev polynomial properties, including generating functions, recursive formulas, orthogonality, and their advantages in interpolation.

Findings

Chebyshev polynomials have important properties like orthogonality and recursive formulas.

They are more accurate than Fourier series for polynomial function approximation.

Chebyshev differential equations are special cases of Sturm-Liouville problems.

Abstract

Ordinary differential equations and boundary value problems arise in many aspects of mathematical physics. Chebyshev differential equation is one special case of the Sturm-Liouville boundary value problem. Generating function, recursive formula, orthogonality, and Parseval's identity are some important properties of Chebyshev polynomials. Compared with a Fourier series, an interpolation function using Chebyshev polynomials is more accurate in approximating polynomial functions. -------- Des \'equations diff\'erentielles ordinaires et des probl\`emes de valeurs limites se posent dans de nombreux aspects de la physique math\'ematique. L'\'equation diff\'erentielle de Chebychev est un cas particulier du probl\`eme de la valeur limite de Sturm-Liouville. La fonction g\'en\'eratrice, la formule r\'ecursive, l'orthogonalit\'e et l'identit\'e de Parseval sont quelques propri\'et\'es importantes du polyn\^ome de Chebyshev. Par rapport \`a une s\'erie de Fourier, une fonction d'interpolation utilisant des polyn\^omes de Chebyshev est plus pr\'ecise dans l'approximation des fonctions polynomiales.