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Albrecht Böttcher, Hermann Brunner, Arieh Iserles, and Syvert P. Nørsett
On the singular values and eigenvalues of the Fox-Li and related operators view print
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| Published: |
December 1, 2010 |
| Keywords: |
Fox-Li operator, Wiener-Hopf operator, oscillatory kernel, eigenvalue, singular value |
| Subject: |
Primary 47B35; Secondary 45C05, 47B05, 65R20, 78A60 |
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Abstract
The Fox-Li operator is a convolution operator over a finite
interval with a special highly oscillatory kernel. It plays an
important role in laser engineering. However, the mathematical
analysis of its spectrum is still rather incomplete. In this expository
paper we survey part of the state of the art, and our emphasis is on
showing how standard Wiener-Hopf theory can be used to obtain
insight into the behaviour of the singular values of the Fox-Li
operator. In addition, several approximations to the spectrum of
the Fox-Li operator are discussed and results on the singular
values and eigenvalues of certain related operators are derived.
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| Acknowledgements
The work of Hermann Brunner was funded by Discovery Grant A9406 of Natural Sciences and Engineering Research Council of Canada.
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| Author information
Albrecht Böttcher:
Fakultät für Mathematik, Technische Universität Chemnitz, 09107 Chemnitz, Germany
aboettch@mathematik.tu-chemnitz.de
Hermann Brunner:
Department of Mathematics and Statistics, Memorial University of Newfoundland, St John's A1C 5S7, Canada
hbrunner@mun.ca
Arieh Iserles:
Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Cambridge CB3 0WA, United Kingdom
A.Iserles@damtp.cam.ac.uk
Syvert P. Nørsett:
Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim 7491, Norway
norsett@math.ntnu.no
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