Discrete Hankel Prolate Spheroidal Wave Functions: Spectral Analysis and Application

TL;DR

This paper introduces discrete Hankel Prolate spheroidal sequences and functions, extending classical spectral analysis results and applying them to estimate bounds related to Ingham's universal constant.

Contribution

It presents a new family of discrete wave functions, DHPSS and DHPSWF, as solutions to a multidimensional energy maximization problem, generalizing Slepian's classical work.

Findings

Defined DHPSS and DHPSWF as eigenfunctions of a finite rank integral operator.

Extended classical decay rate and spectral distribution results.

Provided an upper bound estimate for Ingham's universal constant.

Abstract

Since the early 1960s, the fields of signal processing, data transmission, channel equalisation, filter design and others have been technologically developed and modernised as a result of the research carried out by D. Slepian and his co-authors H. J Landau and H. O Pollack on the time and band-limited wave system known as discrete and continuous spheroidal waves systems. Our aim in this paper is to introduce new discrete wave sequences called discrete Hankel Prolate spheroidal sequences {\bf DHPSS} and their counterparts in the frequency domain called discrete Hankel Prolate spheroidal wave functions {\bf DHPSWF} as radial parts of different solutions of a discrete multidimensional energy maximization problem similar to the one given by D. Slepian and which will generalize his classical pioneering work. In the meantime, we will ensure that our new family is the eigenfunctions set of a finite rank integral operator defined on $L^2(0,\omega),\,0<\omega<1,$ with an associated kernel given by $\sum_{k=1}^N\phi^{\alpha}_{n}(r)\phi^{\alpha}_{n}(r'),$ where $\phi^{\alpha}_{n}(r)=\frac{\sqrt{2r}J_{\alpha}(s_n^{(\alpha)}r)}{|J_{\alpha+1}(s^{(\alpha)}_n)|},0\leq r\leq 1.$ Here $J_\alpha$ is the Bessel function of the first kind and $(s_n^{(\alpha)})_n$ are the associated positive zeros. In addition, we will extend the various classical results proposed concerning the decay rate and spectral distribution associated with the classical case, then we will finish our work by an application on the Ingham's universal constant which we will specify with an upper bound estimate.