Abstract
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 language | English (US) |
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Pages (from-to) | 59-83 |
Number of pages | 25 |
Journal | Applied and Computational Harmonic Analysis |
Volume | 45 |
Issue number | 1 |
DOIs | |
State | Published - Jul 2018 |
Keywords
- Eigenvalues
- Integral operator
- Laplacian
- Non-local boundary value problem
- Rayleigh function
ASJC Scopus subject areas
- Applied Mathematics