On the commutative factorization of n x n matrix Weiner-Hopf kernels with distinct eigenvalues

Veitch, Benjamin H. and Abrahams, I. David (2007) On the commutative factorization of n x n matrix Weiner-Hopf kernels with distinct eigenvalues. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 463 (2078). pp. 613-639. ISSN 1471-2946

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Abstract

In this article, we present a method for factorizing n x n matrix Wiener–Hopf kernels where n > 2 and the factors commute. We are motivated by a method posed by Jones (Jones 1984a Proc. R. Soc. A 393, 185–192) to tackle a narrower class of matrix kernels; however, no matrix of Jones’ form has yet been found to arise in physical Wiener–Hopf models. In contrast, the technique proposed herein should find broad application. To illustrate the approach, we consider a 3 x 3 matrix kernel arising in a problem from elastostatics. While this kernel is not of Jones’ form, we shall show how it can be factorized commutatively. We discuss the essential difference between our method and that of Jones and explain why our method is a generalization. The majority of Wiener–Hopf kernels that occur in canonical diffraction problems are, however, strictly non-commutative. For 2x2 matrices, Abrahams has shown that one can overcome this difficulty using Padé approximants to rearrange a non-commutative kernel into a partial-commutative form; an approximate factorization can then be derived. By considering the dynamic analogue of Antipov’s model, we show for the first time that Abrahams’ Padé approximant method can also be employed within a 3x3 commutative matrix form.

Item Type: Article
Uncontrolled Keywords: Wiener–Hopf technique; matrix Wiener–Hopf; elasticity; elastodynamics; scattering
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 30 Functions of a complex variable
MSC 2010, the AMS's Mathematics Subject Classification > 43 Abstract harmonic analysis
MSC 2010, the AMS's Mathematics Subject Classification > 46 Functional analysis
MSC 2010, the AMS's Mathematics Subject Classification > 47 Operator theory
MSC 2010, the AMS's Mathematics Subject Classification > 78 Optics, electromagnetic theory
MSC 2010, the AMS's Mathematics Subject Classification > 81 Quantum theory
Depositing User: Ms Lucy van Russelt
Date Deposited: 15 Nov 2007
Last Modified: 20 Oct 2017 14:12
URI: http://eprints.maths.manchester.ac.uk/id/eprint/883

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