The reproducing kernel structure associated to Fourier type systems and their quantum analogues

Abstract

We study mapping properties of operators with kernels defined via an abstract formulation of quantum (q-) Fourier type systems. We prove Ismail´s conjecture regarding the existence of a reproducing kernel structure behind these kernels. The results are illustrated with Fourier kernels with ultraspherical and Jacobi weights, their continuous q-extensions and generalizations. As a byproduct of this approach, a new class of sampling theorems is obtained, as well as Neumann type expansions in Bessel and q-Bessel functions.