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The Eigenfunctions of the Hilbert Matrix

Aleman, Alexandru LU ; Montes-Rodriguez, Alfonso and Sarafoleanu, Andreea (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.
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author
organization
publishing date
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
2013-01-30 14:46:08
date last changed
2017-10-29 03:15:10
@article{f30bc9e5-8d36-4199-acd6-9219a2d7fbb3,
  abstract     = {For each noninteger complex number lambda, the Hilbert matrix H-lambda = (1/n+m+lambda)(n,m &gt;= 0) defines a bounded linear operator on the Hardy spaces H-p, 1 &lt; p &lt; a, and on the Korenblum spaces , A(-tau), tau &gt; 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},
  keyword      = {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},
  volume       = {36},
  year         = {2012},
}