2006/01/09 by Luís Daniel Abreu, Abreu, Luis Daniel
Engineering · Mathematics · #33C45 #33D45 #42C15 #44A20 #94A20 #Advanced Numerical Analysis Techniques #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Mathematical Analysis and Transform Methods #Mathematical Physics (math-ph) #Mathematical functions and polynomials
paper · pdf · doi:10.48550/arxiv.math/0601190
openalex publication_date 2006/01/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.