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

Hermann BRUNNER, Arieh Iserles*, Syvert P. Nørsett

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

19 Citations (Scopus)

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.

Original languageEnglish
Pages (from-to)108-130
Number of pages23
JournalIMA Journal of Numerical Analysis
Volume30
Issue number1
DOIs
Publication statusPublished - 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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