Spectral theory of large Wiener–Hopf operators with complex-symmetric kernels and rational symbols

Abstract

Link to this article: http://journals.cambridge.org/abstract_S0305004111000259 How to cite this article: ALBRECHT BÖTTCHER, SERGEI GRUDSKY and ARIEH ISERLES (2011). Spectral theory of large Wiener-Hopf operators with complex-symmetric kernels and rational symbols. Mathematical AbstractThis paper is devoted to the asymptotic behaviour of individual eigenvalues of truncated Wiener-Hopf integral operators over increasing intervals. The kernel of the operators is complex-symmetric and has a rational Fourier transform. Under additional hypotheses, the main result describes the location of the eigenvalues and provides their asymptotic expansions in terms of the reciprocal of the length of the truncation interval. Also determined is the structure of the eigenfunctions.

The Wiener–Hopf technique, its generalizations and applications: constructive and approximate methods

“…A different direction is when the kernel (1.1) has support on the half-line but is not of convolution type (the equation is referred to as being of Hammerstein type) [187]. There is also a wealth of literature on spectral properties of Toeplitz matrices/operators [188–191] which is a related topic but outside the scope of this review.…”

The Wiener--Hopf Technique, its Generalisations and Applications: Constructive and Approximate Methods

“…A different direction is when the kernel (1) has support on the half-line but is not of convolution type (the equation is referred to as being of Hammerstein type) [63]. There is also a wealth of literature on spectral properties of Toeplitz matrices/operators [38,39,68,86] which is a related topic but outside the scope of this review.…”

“…Another interesting discrete system which has an analytic solution via a Wiener-Hopf technique is (38) where |r| = 1. With the help of a Z-transform this can be reduced to a boundary value problem of Carleman type [78].…”

Asymptotics of eigenvalues of large symmetric Toeplitz matrices with smooth simple-loop symbols

“…We note that articles on the individual asymptotics of all eigenvalues have appeared quite recently. Thus, cases of real-valued symbols (self-adjoint Toeplitz operators) satisfying the so-called SL (Simple-Loop) condition were studied in the articles [19], [5], [20]. In these articles, the cases of polynomial, infinitely smooth, having 4 continuous derivatives of symbols were successively considered.…”