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Self-adjoint difference operators and classical solutions to the Stieltjes–Wigert moment problem

descriptionPublicationkeyboard_double_arrow_right Article , Preprint 01 May 2006Embargo end date: 01 Jan 2005Publisher:Elsevier BVJournal:Journal of Approximation Theory, volume 140, pages 1-26 (issn: 0021-9045,

Authors: Jacob S. Christiansen; Erik Koelink;

doi: 10.1016/j.jat.2005.11.010 , 10.48550/arxiv.math/0504456

arXiv: http://arxiv.org/abs/math/0504456

Self-adjoint difference operators and classical solutions to the Stieltjes–Wigert moment problem

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Abstract

The Stieltjes-Wigert polynomials, which correspond to an indeterminate moment problem on the positive half-line, are eigenfunctions of a second order q-difference operator. We consider the orthogonality measures for which the difference operator is symmetric in the corresponding weighted $L^2$-spaces. These measures are exactly the solutions to the q-Pearson equation.In the case of discrete and absolutely continuous measures the difference operator is essentially self-adjoint, and the corresponding spectral decomposition is given explicitly. In particular, we find an orthogonal set of q-Bessel functions complementing the Stieltjes-Wigert polynomials to an orthogonal basis for $L^2(��)$ when $��$ is a discrete orthogonality measure solving the q-Pearson equation. To obtain the spectral decomposition of the difference operator in case of an absolutely continuous orthogonality measure we use the results from the discrete case combined with direct integral techniques.

22 pages; section 2 rewritten, to appear in Journal of Approximation Theory

Related Organizations

KU Leuven
Belgium
Katholieke Universiteit Leuven
Belgium
Delft University of Technology
Netherlands

Keywords

Mathematics(all), Numerical Analysis, Direct integrals of Hilbert spaces, Stieltjes–Wigert polynomials, Applied Mathematics, 47B36 44A60, Spectral analysis, Self-adjoint operators, Functional Analysis (math.FA), Mathematics - Functional Analysis, Mathematics - Classical Analysis and ODEs, Classical Analysis and ODEs (math.CA), FOS: Mathematics, Difference operators, Analysis

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