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On the non-commutative factorization of WienerHopf kernels of Khrapkov type

Summary: On the non-commutative factorization of
Wiener­Hopf kernels of Khrapkov type
B y I. David Abrahams
Department of Mathematics, Keele University,
Keele, Staffordshire ST5 5BG, UK
Received 28 May 1997; revised 22 August 1997; accepted 22 October 1997
There are a great many problems in mathematical physics and engineering which,
when modelled mathematically, are reduced to Wiener­Hopf functional equations
defined in some region of a complex plane. In simple models this equation is scalar,
but in more complicated situations inherent coupling, etc., can often give rise to
coupled systems of equations, and so the resulting Wiener­Hopf kernels are of matrix
form. The crucial step in the solution of any Wiener­Hopf equation is to decompose
the kernel, which can be a quite general function of the complex transform variable,
into a product of two factors which are regular in overlapping half-planes. This can
be accomplished explicitly for scalar kernels, and may be generalized to a particular
class of matrix kernels, namely those which permit a commutative decomposition.
Although this class is wide, and therefore contains some valuable physical problems,
it is found that many such kernels yield factorization elements with exponentially
large behaviour at infinity in their half-planes of analyticity. This prevents a later
step in the Wiener­Hopf procedure being carried through, and therefore introduces


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


Collections: Mathematics