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arXiv:math/0601190 [math.CA]AbstractReferencesReviewsResources

The reproducing kernel structure arising from a combination of continuous and discrete orthogonal polynomials into Fourier systems

Luis Daniel Abreu

Published 2006-01-09, updated 2007-01-05Version 3

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.

Comments: 16 pages; Title changed, major reformulations. To appear in Constr. Approx
Categories: math.CA, math-ph, math.MP
Subjects: 42C15, 44A20, 33C45, 33D45, 94A20
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