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The Eigenfunctions of the Hilbert Matrix

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Author

Summary, in English

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.

Publishing year

2012

Language

English

Pages

353-374

Publication/Series

Constructive Approximation

Volume

36

Issue

3

Document type

Journal article

Publisher

Springer

Topic

  • Mathematical Sciences

Keywords

  • Hilbert matrix
  • Integral operator
  • Eingenvalues
  • Eigenfunctions
  • Differential operators
  • Hypergeometric function
  • Associated Legendre
  • functions of the first kind

Status

Published

ISBN/ISSN/Other

  • ISSN: 0176-4276