### 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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Journal | Applied and Computational Harmonic Analysis |

DOIs | |

State | Accepted/In press - Mar 7 2016 |

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### Keywords

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

### ASJC Scopus subject areas

- Applied Mathematics

### Cite this

**On Rayleigh-type formulas for a non-local boundary value problem associated with an integral operator commuting with the Laplacian.** / Hermi, Lotfi; Saito, Naoki.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - On Rayleigh-type formulas for a non-local boundary value problem associated with an integral operator commuting with the Laplacian

AU - Hermi, Lotfi

AU - Saito, Naoki

PY - 2016/3/7

Y1 - 2016/3/7

N2 - 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.

AB - 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.

KW - Eigenvalues

KW - Integral operator

KW - Laplacian

KW - Non-local boundary value problem

KW - Rayleigh function

UR - http://www.scopus.com/inward/record.url?scp=85005942706&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85005942706&partnerID=8YFLogxK

U2 - 10.1016/j.acha.2016.08.003

DO - 10.1016/j.acha.2016.08.003

M3 - Article

AN - SCOPUS:85005942706

JO - Applied and Computational Harmonic Analysis

JF - Applied and Computational Harmonic Analysis

SN - 1063-5203

ER -