TY - JOUR
T1 - The spectral problem for a class of highly oscillatory Fredholm integral operators
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.
PY - 2010/1
Y1 - 2010/1
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
KW - Complex-symmetric Fredholm integral operator
KW - Fredholm integral equations
KW - Highly oscillatory kernel
UR - http://www.scopus.com/inward/record.url?scp=76549100247&partnerID=8YFLogxK
U2 - 10.1093/imanum/drn060
DO - 10.1093/imanum/drn060
M3 - Journal article
AN - SCOPUS:76549100247
SN - 0272-4979
VL - 30
SP - 108
EP - 130
JO - IMA Journal of Numerical Analysis
JF - IMA Journal of Numerical Analysis
IS - 1
ER -