Prolate Spheroidal Wave Functions In q-Fourier Analysis

TL;DR

This paper explores the properties of prolate spheroidal wave functions within q-Fourier analysis, revealing new systems with unique orthogonality features and introducing a novel q-based sampling formula.

Contribution

It introduces new orthogonal systems in q-Fourier analysis that share properties with classical spheroidal wave functions and presents a new sampling formula using q^n points.

Findings

Existence of new orthogonal systems in q-Fourier analysis.

Development of a q-based sampling formula with q^n points.

Extension of classical spheroidal wave function properties to q-analogs.

Abstract

The prolate spheroidal wave functions, which are a special case of the spheroidal wave functions, possess a very surprising and unique property [6]. They are an orthogonal basis of both $L^2(-1,1)$ and the Paley-Wiener space of bandlimited functions. They also satisfy a discrete orthogonality relation. No other system of classical orthogonal functions is known to possess this strange property. We prove that there are new systems possessing this property in $q$-Fourier analysis. As application we give a new sampling formula with $q^n$ as sampling points, where 0 < q < 1.