On Rayleigh-type formulas for a non-local boundary value problem associated with an integral operator commuting with the Laplacian

Lotfi Hermi, Naoki Saito

Research output: Contribution to journalArticle


In this article we prove the existence, uniqueness, and simplicity of a negative eigenvalue for a class of integral operators whose kernel is of the form . |x-y|ρ, . 0<ρ≤1, . x,y∈[-a,a]. We also provide two different ways of producing recursive formulas for the Rayleigh functions (i.e., recursion formulas for power sums) of the eigenvalues of this integral operator when . ρ=1, providing means of approximating this negative eigenvalue. These methods offer recursive procedures for dealing with the eigenvalues of a one-dimensional Laplacian with non-local boundary conditions which commutes with an integral operator having a harmonic kernel. The problem emerged in recent work by one of the authors [48]. We also discuss extensions in higher dimensions and links with distance matrices.

Original languageEnglish (US)
JournalApplied and Computational Harmonic Analysis
Publication statusAccepted/In press - Mar 7 2016



  • Eigenvalues
  • Integral operator
  • Laplacian
  • Non-local boundary value problem
  • Rayleigh function

ASJC Scopus subject areas

  • Applied Mathematics

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