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

Bottcher, Albrecht and Brunner, Hermann and Iserles, Arieh and Norsett, Syver P. (2010) On the singular values and eigenvalues of the Fox–Li and related operators. New York Journal of Mathematics , 16. pp. 539-561. ISSN 1076-9803

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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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