Skip to main content

Lund University Publications

LUND UNIVERSITY LIBRARIES

Self-adjoint difference operators and symmetric Al-Salam-Chihara polynomials

Christiansen, Jacob S. LU and Koelink, Erik (2008) In Constructive Approximation 28(2). p.199-218
Abstract

The symmetric Al-Salam-Chihara polynomials for q > 1 are associated with an indeterminate moment problem. There is a self-adjoint second-order difference operator on 2(Z) to which these polynomials are eigenfunctions. We determine the spectral decomposition of this self-adjoint operator. This leads to a class of discrete orthogonality measures, which have been obtained previously by Christiansen and Ismail using a different method, and we give an explicit orthogonal basis for the corresponding weighted l 2-space. In particular, the orthocomplement of the polynomials is described explicitly. Taking a limit we obtain all the N-extremal solutions to the q-1-Hermite moment problem, a result originally... (More)

The symmetric Al-Salam-Chihara polynomials for q > 1 are associated with an indeterminate moment problem. There is a self-adjoint second-order difference operator on 2(Z) to which these polynomials are eigenfunctions. We determine the spectral decomposition of this self-adjoint operator. This leads to a class of discrete orthogonality measures, which have been obtained previously by Christiansen and Ismail using a different method, and we give an explicit orthogonal basis for the corresponding weighted l 2-space. In particular, the orthocomplement of the polynomials is described explicitly. Taking a limit we obtain all the N-extremal solutions to the q-1-Hermite moment problem, a result originally obtained by Ismail and Masson in a different way. Some applications of the results are discussed.

(Less)
Please use this url to cite or link to this publication:
author
and
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Al-Salam-Chihara polynomials, Denseness of polynomials, Difference operators, Spectral analysis
in
Constructive Approximation
volume
28
issue
2
pages
20 pages
publisher
Springer
external identifiers
  • scopus:38549115746
ISSN
0176-4276
DOI
10.1007/s00365-007-0677-x
language
English
LU publication?
no
id
8329fcf9-fd2f-419b-b841-a37d932381e6
date added to LUP
2025-07-11 11:50:08
date last changed
2025-10-14 10:03:47
@article{8329fcf9-fd2f-419b-b841-a37d932381e6,
  abstract     = {{<p>The symmetric Al-Salam-Chihara polynomials for q &gt; 1 are associated with an indeterminate moment problem. There is a self-adjoint second-order difference operator on <sup>2</sup>(Z) to which these polynomials are eigenfunctions. We determine the spectral decomposition of this self-adjoint operator. This leads to a class of discrete orthogonality measures, which have been obtained previously by Christiansen and Ismail using a different method, and we give an explicit orthogonal basis for the corresponding weighted l <sup>2</sup>-space. In particular, the orthocomplement of the polynomials is described explicitly. Taking a limit we obtain all the N-extremal solutions to the q<sup>-1</sup>-Hermite moment problem, a result originally obtained by Ismail and Masson in a different way. Some applications of the results are discussed.</p>}},
  author       = {{Christiansen, Jacob S. and Koelink, Erik}},
  issn         = {{0176-4276}},
  keywords     = {{Al-Salam-Chihara polynomials; Denseness of polynomials; Difference operators; Spectral analysis}},
  language     = {{eng}},
  number       = {{2}},
  pages        = {{199--218}},
  publisher    = {{Springer}},
  series       = {{Constructive Approximation}},
  title        = {{Self-adjoint difference operators and symmetric Al-Salam-Chihara polynomials}},
  url          = {{http://dx.doi.org/10.1007/s00365-007-0677-x}},
  doi          = {{10.1007/s00365-007-0677-x}},
  volume       = {{28}},
  year         = {{2008}},
}