 
Summary: ON THE SOLUTION OF WIENERHOPF PROBLEMS INVOLVING
NONCOMMUTATIVE MATRIX KERNEL DECOMPOSITIONS
I. DAVID ABRAHAMS
SIAM J. APPL. MATH. c 1997 Society for Industrial and Applied Mathematics
Vol. 57, No. 2, pp. 541567, April 1997 013
Abstract. Many problems in physics and engineering with semiinfinite boundaries or interfaces
are exactly solvable by the WienerHopf technique. It has been used successfully in a multitude of
different disciplines when the WienerHopf functional equation contains a single scalar kernel. For
complex boundary value problems, however, the procedure often leads to coupled equations which
therefore have a kernel of matrix form. The key step in the technique is to decompose the kernel
into a product of two functions, one analytic in an upper region of a complex (transform) plane and
the other analytic in an overlapping lower halfplane. This is straightforward for scalar kernels but
no method has yet been proposed for general matrices.
In this article a new procedure is introduced whereby Pad´e approximants are employed to obtain
an approximate but explicit noncommutative factorization of a matrix kernel. As well as being
simple to apply, the use of approximants allows the accuracy of the factorization to be increased
almost indefinitely. The method is demonstrated by way of example. Scattering of acoustic waves
by a semiinfinite screen at the interface between two compressible media with different physical
properties is examined in detail. Numerical evaluation of the approximate factorizations together
with results on an upper bound of the absolute error reveal that convergence with increasing Pad´e
