The application of Pade approximants 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 v ariable, alpha say, into a product of two terms. One is singularity and zer o free in an upper region of the alpha -plane, and the other singularity an d zero free in an overlapping lower region. Each product factor can be expr essed in terms of a Cauchy-type integral formula, but this form presents di fficulties due to the speed of its evaluation and numerical problems caused by singularities near the integration contour. Other representations are a vailable in special cases, for instance an infinite product form. for merom orphic functions, but not in general. To overcome these problems, several a pproximate methods for decomposing the transformed kernels have been sugges ted. However, whilst these offer simple explicit expressions, their forms t end to have been derived in an ad hoc fashion and to date have only mediocr e accuracy (of order one per cent or so). A new method for approximating Wiener-Hopf kernels is offered in this artic le which employs Pade approximants. These have the advantage of offering ve ry simple approximate factors of Fourier transformed kernels which are foun d to be extremely accurate for modest computational effort. Further, the de rivation of the factors is algorithmic and therefore requires little effort , and the Pade number is a convenient parameter with which to reduce errors to within set target values. The paper demonstrates the efficacy of the ap proach on several model kernels, and numerical results presented herein con firm theoretical predictions regarding convergence to the exact results, et c. The relationship between the present method and earlier approximate sche mes is discussed.