Approximations in Sobolev Spaces by Prolate Spheroidal Wave Functions

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.

arXiv admin note: substantial text overlap with arXiv:1012.3881

References in corpus (1)

• Spectral Decay of Time and Frequency Limiting Operator

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