Approximation of almost time and band limited functions I: Hermite expansions

TL;DR

This paper studies how well almost time and band limited functions can be approximated using Hermite and scaled Hermite expansions, providing convergence rates in Sobolev spaces with fixed support.

Contribution

It introduces new approximation results for almost time and band limited functions using Hermite expansions, including convergence rate analysis in Sobolev spaces.

Findings

Hermite expansions effectively approximate almost time and band limited functions.

Convergence rates are established for Sobolev space functions with fixed support.

Results improve understanding of Hermite expansion efficiency for practical signal approximation.

Abstract

The aim of this paper is to investigate the quality of approximation of almost time and band limited functions by its expansion in the Hermite and scaled Hermite basis. As a corollary, this allows us to obtain the rate of convergence of the Hermite expansion of function in the $L^2$-Sobolev space with fixed compact support.