The Reproducing Kernel Structure Arising from a Combination of Continuous and Discrete Orthogonal Polynomials into Fourier Systems |
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Authors: | Luis Daniel Abreu |
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Institution: | (1) Department of Mathematics, University of Coimbra, Apartado 3008, 3001-454 Coimbra, Portugal;(2) NUHAG, Faculty of Mathematics, University of Vienna, Nordbergstrasse 15, A-1090 Wien, Austria |
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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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Keywords: | |
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