The Eigenfunctions of the Hilbert Matrix
(2012) In Constructive Approximation 36(3). p.353374 Abstract
 For each noninteger complex number lambda, the Hilbert matrix Hlambda = (1/n+m+lambda)(n,m >= 0) defines a bounded linear operator on the Hardy spaces Hp, 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:
http://lup.lub.lu.se/record/3379419
 author
 Aleman, Alexandru ^{LU} ; MontesRodriguez, 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
 01764276
 DOI
 10.1007/s003650129157z
 language
 English
 LU publication?
 yes
 id
 f30bc9e58d364199acd69219a2d7fbb3 (old id 3379419)
 date added to LUP
 20130130 14:46:08
 date last changed
 20180114 03:13:28
@article{f30bc9e58d364199acd69219a2d7fbb3, abstract = {For each noninteger complex number lambda, the Hilbert matrix Hlambda = (1/n+m+lambda)(n,m >= 0) defines a bounded linear operator on the Hardy spaces Hp, 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 MontesRodriguez, Alfonso and Sarafoleanu, Andreea}, issn = {01764276}, keyword = {Hilbert matrix,Integral operator,Eingenvalues,Eigenfunctions,Differential operators,Hypergeometric function,Associated Legendre,functions of the first kind}, language = {eng}, number = {3}, pages = {353374}, publisher = {Springer}, series = {Constructive Approximation}, title = {The Eigenfunctions of the Hilbert Matrix}, url = {http://dx.doi.org/10.1007/s003650129157z}, volume = {36}, year = {2012}, }