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On the commutative factorization of n!n matrix WienerHopf kernels
 

Summary: On the commutative factorization
of n!n matrix Wiener­Hopf kernels
with distinct eigenvalues
BY BENJAMIN H. VEITCH

AND I. DAVID ABRAHAMS*
School of Mathematics, University of Manchester, Oxford Road,
Manchester M13 9PL, UK
In this article, we present a method for factorizing n!n matrix Wiener­Hopf kernels
where nO2 and the factors commute. We are motivated by a method posed by Jones
(Jones 1984a Proc. R. Soc. A 393, 185­192) to tackle a narrower class of matrix kernels;
however, no matrix of Jones' form has yet been found to arise in physical Wiener­Hopf
models. In contrast, the technique proposed herein should find broad application. To
illustrate the approach, we consider a 3!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 Wiener­Hopf kernels that occur in canonical diffraction problems are,
however, strictly non-commutative. For 2!2 matrices, Abrahams has shown that one
can overcome this difficulty using Pade´ approximants to rearrange a non-commutative

  

Source: Abrahams, I. David - Department of Mathematics, University of Manchester

 

Collections: Mathematics