Abderrazek Karoui

Philippe Jaming, Abderrazek Karoui, Susanna Spektor

The aim of this paper is to investigate the quality of approximation of almost time and almost band-limited functions by its expansion in three classical orthogonal polynomials bases: the Hermite, Legendre and Chebyshev bases. As a corollary, this allows us to obtain the quality of approximation in the L 2 --Sobolev space by these orthogonal polynomials bases. Also, we obtain the rate of the Legendre series expansion of the prolate spheroidal wave functions. Some numerical examples are given to illustrate the different results of this work.

Aline Bonami, Abderrazek Karoui

For fixed $c,$ the Prolate Spheroidal Wave Functions (PSWFs) $ψ_{n, c}$ form a basis with remarkable properties for the space of band-limited functions with bandwidth $c$. They have been largely studied and used after the seminal work of D. Slepian, H. Landau and H. Pollack. Recently, they have been used for the approximation of functions in the Sobolev space $H^s([-1,1])$. In view of this, we give new estimates on the decay rate of eigenvalues of the Sinc kernel integral operators. This is one of the main issues of this work. A second one is the choice of the parameter $c$ when approximating a function in $H^s([-1,1])$ by its truncated PSWFs series expansion. Such functions may be seen as the restriction to $[-1,1]$ of almost time-limited and band-limited functions, for which PSWFs expansions are still well adapted. Finally, we provide the reader with some numerical examples that illustrate the different results of this work.

Philippe Jaming, Abderrazek Karoui, Ron Kerman et al.

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.