Approximations in Sobolev spaces by prolate spheroidal wave functions

• Published in: Applied and computational harmonic analysis Vol. 42; no. 3; pp. 361 - 377
• Main Authors: Bonami, Aline, Karoui, Abderrazek
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.05.2017; Elsevier
• Subjects: Classical Analysis and ODEs; Eigenvalues and eigenfunctions estimates; Mathematics; Prolate spheroidal wave functions; Sobolev spaces; Spectral approximation; secondary; Eigenvalues and eigenfunctions estimates; Spectral approximation; Sobolev spaces; Prolate spheroidal wave functions; primary; eigenvalues and eigenfunctions estimates; spectral approximation; 65L15 Key words and phrases Prolate spheroidal wave functions; 2010 Mathematics Subject Classification Primary 42C10; 65L70 Secondary 41A60
• ISSN: 1063-5203; 1096-603X
• DOI: 10.1016/j.acha.2015.09.001

Recently, there is a growing interest in the spectral approximation by the Prolate Spheroidal Wave Functions (PSWFs) ψn,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 Hs([−1,1]). The quality of the spectral approximation and the choice of the parameter c when approximating a function in Hs([−1,1]) by its truncated PSWFs series expansion, are the main issues. By considering a function f∈Hs([−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.