The approximation of almost time and band limited functions by their expansion in some orthogonal polynomials bases

TL;DR

This paper analyzes how well almost time and band-limited functions can be approximated using Hermite, Legendre, and Chebyshev polynomial bases, providing error estimates and approximation rates.

Contribution

It offers new bounds on approximation quality in L2-Sobolev spaces and determines the Legendre series expansion rate for prolate spheroidal wave functions.

Findings

Approximation errors are quantified for Hermite, Legendre, and Chebyshev bases.

Derived the rate of Legendre series expansion for prolate spheroidal wave functions.

Numerical examples validate theoretical approximation bounds.

Abstract

The aim of this paper is to investigate the quality of approximation of almost time and almost band-limited functions by its expansion in three classical orthogonal polynomials bases: the Hermite, Legendre and Chebyshev bases. As a corollary, this allows us to obtain the quality of approximation in the L 2 --Sobolev space by these orthogonal polynomials bases. Also, we obtain the rate of the Legendre series expansion of the prolate spheroidal wave functions. Some numerical examples are given to illustrate the different results of this work.