The application of Padéapproximants 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, α say, into a product of two terms. One is singularity and zero free in an upper region of the α-plane, and the other singularity and zero free in an overlapping lower region. Each product factor can be expressed in terms of a Cauchy-type integral formula, but this form presents difficulties due to the speed of its evaluation and numerical problems caused by singularities near the integration contour. 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 approximate methods for decomposing the transformed kernels have been suggested. However, whilst these offer simple explicit expressions, their forms tend to have been derived in an ad hoc fashion and to date have only mediocre accuracy (of order one per cent or so). A new method for approximating Wiener–Hopf kernels is offered in this article which employs Padéapproximants. These have the advantage of offering very simple approximate factors of Fourier transformed kernels which are found to be extremely accurate for modest computational effort. Further, the derivation of the factors is algorithmic and therefore requires little effort, and the Padénumber is a convenient parameter with which to reduce errors to within set target values. The paper demonstrates the efficacy of the approach on several model kernels, and numerical results presented herein confirm theoretical predictions regarding convergence to the exact results, etc. The relationship between the present method and earlier approximate schemes is discussed.