The application of Padeapproximants to Wiener-Hopf factorization

The key step in the solution of a Wiener–Hopf equation is the decomposition of the Fourier transform of the kernel,which is a function of a complex variable, say, into a productof two terms. One is singularity and zero free in an upperregion of the -plane, and the other singularity and zero freein an overlapping lower region. Each product factor can beexpressed in terms of a Cauchy-type integral formula, but thisform presents difficulties due to the speed of its evaluationand numerical problems caused by singularities near the integrationcontour. Other representations are available in special cases,for instance an infinite product form for meromorphic functions,but not in general. To overcome these problems, several approximatemethods for decomposing the transformed kernels have been suggested.However, whilst these offer simple explicit expressions, theirforms tend to have been derived in an ad hoc fashion and todate have only mediocre accuracy (of order one per cent orso). A new method for approximating Wiener–Hopf kernelsis offered in this article which employs Padéapproximants.These have the advantage of offering very simple approximatefactors of Fourier transformed kernels which are found to beextremely accurate for modest computational effort. Further,the derivation of the factors is algorithmic and thereforerequires little effort, and the Padénumber is a convenientparameter with which to reduce errors to within set targetvalues. The paper demonstrates the efficacy of the approachon several model kernels, and numerical results presented hereinconfirm theoretical predictions regarding convergence to theexact results, etc. The relationship between the present methodand earlier approximate schemes is discussed. Received 7 February, 1998. Revised 18 February, 1999.