The spectral problem for a class of highly oscillatory Fredholm integral operators

Hermann BRUNNER; Arieh Iserles; Syvert P. Nørsett

doi:10.1093/imanum/drn060

The spectral problem for a class of highly oscillatory Fredholm integral operators

Hermann BRUNNER, Arieh Iserles^*, Syvert P. Nørsett

^*Corresponding author for this work

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

26 Citations (Scopus)

Abstract

Let ℱ_ω be a linear, complex-symmetric Fredholm integral operator with highly oscillatory kernel K₀(x, y)e ^iωx-y. We study the spectral problem for large ω, showing that the spectrum consists of infinitely many discrete (complex) eigenvalues and give a precise description of the way in which they converge to the origin. In addition, we investigate the asymptotic properties of the solutions f = f(x;ω) to the associated Fredholm integral equation f = μℱ_ωf + a as ω→∞, thus refining a classical result by Ursell. Possible extensions of these results to highly oscillatory Fredholm integral operators with more general highly oscillating kernels are also discussed.

Original language	English
Pages (from-to)	108-130
Number of pages	23
Journal	IMA Journal of Numerical Analysis
Volume	30
Issue number	1
DOIs	https://doi.org/10.1093/imanum/drn060
Publication status	Published - Jan 2010

Scopus Subject Areas

Mathematics(all)
Computational Mathematics
Applied Mathematics

User-Defined Keywords

Asymptotic behaviour of highly oscillatory solutions
Asymptotic behaviour of spectrum
Complex-symmetric Fredholm integral operator
Fredholm integral equations
Highly oscillatory kernel

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10.1093/imanum/drn060

Cite this

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abstract = "Let ℱω be a linear, complex-symmetric Fredholm integral operator with highly oscillatory kernel K0(x, y)e iωx-y. We study the spectral problem for large ω, showing that the spectrum consists of infinitely many discrete (complex) eigenvalues and give a precise description of the way in which they converge to the origin. In addition, we investigate the asymptotic properties of the solutions f = f(x;ω) to the associated Fredholm integral equation f = μℱωf + a as ω→∞, thus refining a classical result by Ursell. Possible extensions of these results to highly oscillatory Fredholm integral operators with more general highly oscillating kernels are also discussed.",

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author = "Hermann BRUNNER and Arieh Iserles and N{\o}rsett, {Syvert P.}",

note = "Funding Information: Natural Sciences and Engineering Research Council of Canada (NSERC Discovery Grant No. 9406) to H.B.",

year = "2010",

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doi = "10.1093/imanum/drn060",

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AU - BRUNNER, Hermann

AU - Iserles, Arieh

AU - Nørsett, Syvert P.

N1 - Funding Information: Natural Sciences and Engineering Research Council of Canada (NSERC Discovery Grant No. 9406) to H.B.

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N2 - Let ℱω be a linear, complex-symmetric Fredholm integral operator with highly oscillatory kernel K0(x, y)e iωx-y. We study the spectral problem for large ω, showing that the spectrum consists of infinitely many discrete (complex) eigenvalues and give a precise description of the way in which they converge to the origin. In addition, we investigate the asymptotic properties of the solutions f = f(x;ω) to the associated Fredholm integral equation f = μℱωf + a as ω→∞, thus refining a classical result by Ursell. Possible extensions of these results to highly oscillatory Fredholm integral operators with more general highly oscillating kernels are also discussed.

AB - Let ℱω be a linear, complex-symmetric Fredholm integral operator with highly oscillatory kernel K0(x, y)e iωx-y. We study the spectral problem for large ω, showing that the spectrum consists of infinitely many discrete (complex) eigenvalues and give a precise description of the way in which they converge to the origin. In addition, we investigate the asymptotic properties of the solutions f = f(x;ω) to the associated Fredholm integral equation f = μℱωf + a as ω→∞, thus refining a classical result by Ursell. Possible extensions of these results to highly oscillatory Fredholm integral operators with more general highly oscillating kernels are also discussed.

KW - Asymptotic behaviour of highly oscillatory solutions

KW - Asymptotic behaviour of spectrum

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The spectral problem for a class of highly oscillatory Fredholm integral operators

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