Chromatic derivatives, chromatic expansions and associated spaces II

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.