Chromatic derivatives, chromatic expansions and associated spaces II

TL;DR

This paper explores differential operators linked to orthogonal polynomial families, leading to basis functions for function expansions, revealing new identities, and introducing novel almost periodic function spaces with applications in signal processing.

Contribution

It introduces new differential operators and function spaces associated with orthogonal polynomials, extending classical special functions and their identities, with practical applications in signal recovery.

Findings

Basis functions include classical special functions like Bessel functions.

New spaces of almost periodic functions are constructed.

Applications demonstrated in non-uniform signal sampling and recovery.

Abstract

We study differential operators associated with families of polynomials orthonormal with respect to certain measures. These operators, when applied to the Fourier transforms of such measures, produce basis functions for expansions of functions analytic on some complex domains. For many classical families of orthogonal polynomials these basis functions are the familiar special functions, such as the Bessel and the spherical Bessel functions. Many familiar identities involving such special functions turn out to be just special cases of such expansions. We also use these differential operators to introduce some new spaces of almost periodic functions. The notions we study here have been successfully applied to signal processing, for example to recovery of band-limited signals from their non-uniform samples as well as from their zero crossings and the locations of their extremal points.