Prolate Spheroidal Wave Functions In q-Fourier Analysis

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.