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

Christiansen, Jacob S. LU and Koelink, Erik (2006) In Journal of Approximation Theory 140(1). p.1-26
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 L2-spaces. Under some additional assumptions 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 L2 ( μ... (More)

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 L2-spaces. Under some additional assumptions 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 L2 ( μ ) when μ 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.

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author
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publishing date
type
Contribution to journal
publication status
published
subject
keywords
Difference operators, Direct integrals of Hilbert spaces, Self-adjoint operators, Spectral analysis, Stieltjes-Wigert polynomials
in
Journal of Approximation Theory
volume
140
issue
1
pages
26 pages
publisher
Academic Press
external identifiers
  • scopus:33646377394
ISSN
0021-9045
DOI
10.1016/j.jat.2005.11.010
language
English
LU publication?
no
id
300bbe34-e9eb-4298-94a7-62cdf5207248
date added to LUP
2025-07-11 12:20:32
date last changed
2025-10-14 10:24:22
@article{300bbe34-e9eb-4298-94a7-62cdf5207248,
  abstract     = {{<p>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<sup>2</sup>-spaces. Under some additional assumptions 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<sup>2</sup> ( μ ) when μ 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.</p>}},
  author       = {{Christiansen, Jacob S. and Koelink, Erik}},
  issn         = {{0021-9045}},
  keywords     = {{Difference operators; Direct integrals of Hilbert spaces; Self-adjoint operators; Spectral analysis; Stieltjes-Wigert polynomials}},
  language     = {{eng}},
  number       = {{1}},
  pages        = {{1--26}},
  publisher    = {{Academic Press}},
  series       = {{Journal of Approximation Theory}},
  title        = {{Self-adjoint difference operators and classical solutions to the Stieltjes-Wigert moment problem}},
  url          = {{http://dx.doi.org/10.1016/j.jat.2005.11.010}},
  doi          = {{10.1016/j.jat.2005.11.010}},
  volume       = {{140}},
  year         = {{2006}},
}