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Approximations in Sobolev Spaces by Prolate Spheroidal Wave Functions

Authors
  • Bonami, Aline
  • Karoui, Abderrazek
Type
Preprint
Publication Date
Sep 09, 2015
Submission Date
Sep 09, 2015
Identifiers
DOI: 10.1016/j.acha.2015.09.001
Source
arXiv
License
Yellow
External links

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.

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