The Eigenfunctions of the Hilbert Matrix
(2012) In Constructive Approximation 36(3). p.353-374- Abstract
- For each noninteger complex number lambda, the Hilbert matrix H-lambda = (1/n+m+lambda)(n,m >= 0) defines a bounded linear operator on the Hardy spaces H-p, 1 < p < a, and on the Korenblum spaces , A(-tau), tau > 0. In this work, we determine the point spectrum with multiplicities of the Hilbert matrix acting on these spaces. This extends to complex lambda results by Hill and Rosenblum for real lambda. We also provide a closed formula for the eigenfunctions. They are in fact closely related to the associated Legendre functions of the first kind. The results will be achieved through the analysis of certain differential operators in the commutator of the Hilbert matrix.
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/3379419
- author
- Aleman, Alexandru LU ; Montes-Rodriguez, Alfonso and Sarafoleanu, Andreea
- organization
- publishing date
- 2012
- type
- Contribution to journal
- publication status
- published
- subject
- keywords
- Hilbert matrix, Integral operator, Eingenvalues, Eigenfunctions, Differential operators, Hypergeometric function, Associated Legendre, functions of the first kind
- in
- Constructive Approximation
- volume
- 36
- issue
- 3
- pages
- 353 - 374
- publisher
- Springer
- external identifiers
-
- wos:000311363400002
- scopus:84869887494
- ISSN
- 0176-4276
- DOI
- 10.1007/s00365-012-9157-z
- language
- English
- LU publication?
- yes
- id
- f30bc9e5-8d36-4199-acd6-9219a2d7fbb3 (old id 3379419)
- date added to LUP
- 2016-04-01 10:42:07
- date last changed
- 2022-01-26 01:41:27
@article{f30bc9e5-8d36-4199-acd6-9219a2d7fbb3, abstract = {{For each noninteger complex number lambda, the Hilbert matrix H-lambda = (1/n+m+lambda)(n,m >= 0) defines a bounded linear operator on the Hardy spaces H-p, 1 < p < a, and on the Korenblum spaces , A(-tau), tau > 0. In this work, we determine the point spectrum with multiplicities of the Hilbert matrix acting on these spaces. This extends to complex lambda results by Hill and Rosenblum for real lambda. We also provide a closed formula for the eigenfunctions. They are in fact closely related to the associated Legendre functions of the first kind. The results will be achieved through the analysis of certain differential operators in the commutator of the Hilbert matrix.}}, author = {{Aleman, Alexandru and Montes-Rodriguez, Alfonso and Sarafoleanu, Andreea}}, issn = {{0176-4276}}, keywords = {{Hilbert matrix; Integral operator; Eingenvalues; Eigenfunctions; Differential operators; Hypergeometric function; Associated Legendre; functions of the first kind}}, language = {{eng}}, number = {{3}}, pages = {{353--374}}, publisher = {{Springer}}, series = {{Constructive Approximation}}, title = {{The Eigenfunctions of the Hilbert Matrix}}, url = {{http://dx.doi.org/10.1007/s00365-012-9157-z}}, doi = {{10.1007/s00365-012-9157-z}}, volume = {{36}}, year = {{2012}}, }