On the commutative factorization of nxn matrix Wiener-Hopf kernels with distinct eigenvalues

In this article, we present a method for factorizing nxn 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. A393, 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 3x3 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. © 2006 The Royal Society.