Approximations in Sobolev Spaces by Prolate Spheroidal Wave Functions

TL;DR

This paper investigates the spectral approximation of functions in Sobolev spaces using Prolate Spheroidal Wave Functions, analyzing the approximation quality and parameter selection with theoretical insights and numerical examples.

Contribution

It provides new analysis on the approximation properties of PSWFs in Sobolev spaces and offers guidance on choosing parameters for optimal spectral approximation.

Findings

PSWFs form a basis for Sobolev spaces with good approximation properties.

Optimal parameter c depends on the function's smoothness and bandwidth.

Numerical examples illustrate the effectiveness of the approximation methods.

Abstract

Recently, there is a growing interest in the spectral approximation by the Prolate Spheroidal Wave Functions (PSWFs) $\psi_{n, c},\, c>0.$ This is due to the promising new contributions of these functions in various classical as well as emerging applications from Signal Processing, Geophysics, Numerical Analysis, etc. The PSWFs form a basis with remarkable properties not only for the space of band-limited functions with bandwidth $c,$ but also for the Sobolev space $H^s([-1,1])$. The quality of the spectral approximation and the choice of the parameter $c$ when approximating a function in $H^s([-1,1])$ by its truncated PSWFs series expansion, are the main issues. By considering a function $f\in H^s([-1,1])$ as the restriction to $[-1,1]$ of an almost time-limited and band-limited function, we try to give satisfactory answers to these two issues. Also, we illustrate the different results of this work by some numerical examples.