New York Journal of Mathematics
Volume 16 (2010) 539-561


Albrecht Böttcher, Hermann Brunner, Arieh Iserles, and Syvert P. Nørsett

On the singular values and eigenvalues of the Fox-Li and related operators

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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

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.


The work of Hermann Brunner was funded by Discovery Grant A9406 of Natural Sciences and Engineering Research Council of Canada.

Author information

Albrecht Böttcher:
Fakultät für Mathematik, Technische Universität Chemnitz, 09107 Chemnitz, Germany

Hermann Brunner:
Department of Mathematics and Statistics, Memorial University of Newfoundland, St John's A1C 5S7, Canada

Arieh Iserles:
Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Cambridge CB3 0WA, United Kingdom

Syvert P. Nørsett:
Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim 7491, Norway