Abstract
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’s conjecture regarding the existence of a reproducing kernel structure behind these kernels, by establishing a link with Saitoh’s theory of linear transformations in Hilbert space. The results are illustrated with Fourier kernels with ultraspherical, 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.
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Abreu, L. The Reproducing Kernel Structure Arising from a Combination of Continuous and Discrete Orthogonal Polynomials into Fourier Systems. Constr Approx 28, 219–235 (2008). https://doi.org/10.1007/s00365-006-0657-0
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DOI: https://doi.org/10.1007/s00365-006-0657-0