Indexed on: 05 Jan '07Published on: 05 Jan '07Published in: Mathematics - Classical Analysis and ODEs
We study mapping properties of operators with kernels defined via a combination of continuous and discrete orthogonal polynomials, which provide an abstract formulation of quantum (q-) Fourier type systems. We prove Ismail conjecture regarding the existence of a reproducing kernel structure behind these kernels, by establishing a link with Saitoh theory of linear transformations in Hilbert space. The results are illustrated with Fourier kernels with ultraspherical 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.