Abderrazek Karoui

Spectral analysis of the finite Hankel transform and circular prolate spheroidal wave functions

In this paper, we develop two practical methods for the computation of the eigenvalues as well as the eigenfunctions of the finite Hankel transform operator. These different eigenfunctions are called circular prolate spheroidal wave functions (CPSWFs). This work is motivated by the potential applications of the CPSWFs as well as the development of practical methods for computing their values. Also, in this work, we should prove that the CPSWFs form an orthonormal basis of the space of Hankel band-limited functions, an orthogonal basis of L^2([0,1]) and an orthonormal system of L^2([0,+~[). Our computation of the CPSWFs and their associated eigenvalues is done by the use of two different methods. The first method is based on a suitable matrix representation of the finite Hankel transform operator. The second method is based on the use of an efficient quadrature method based on a special family of orthogonal polynomials. Also, we give two Maple programs that implement the previous two methods. Finally, we present some numerical results that illustrate the results of this work.

Uniform Approximation and Explicit Estimates for the Prolate Spheroidal Wave Functions

For fixed c, prolate spheroidal wave functions (PSWFs), denoted by

A note on the design of nonseparable orthonormal wavelet bases of L2(R3)

\psi _{n, c}

Generalized Prolate Spheroidal Wave Functions: Spectral Analysis and Approximation of Almost Band-Limited Functions

ψn,c, form an orthogonal basis with remarkable properties for the space of band-limited functions with bandwith c. They have been widely studied and used after the seminal work of D. Slepian and his co-authors. In several applications, uniform estimates of the

Uncertainty Principles, Prolate Spheroidal Wave Functions, and Applications

\psi _{n,c}

The approximation of almost time- and band-limited functions by their expansion in some orthogonal polynomials bases

ψn,c in n and c are needed. To progress in this direction, we push forward the uniform approximation error bounds and give an explicit approximation of their values at 1 in terms of the Legendre complete elliptic integral of the first kind. Also, we give an explicit formula for the accurate approximation of the eigenvalues of the Sturm–Liouville operator associated with the PSWFs.