Reinvigorating the Wiener-Hopf technique in the pursuit of understanding processes and materials

Abstract

This perspective originated during the Isaac Newton Institute for Mathematical Sciences research programme “Bringing pure and applied analysis together via the Wiener-Hopf technique, its generalisations and applications (WHT)”. It fuelled intensive discussions on current pure and applied mathematical challenges of the Wiener-Hopf technique, driven by its impact on real-world problems.

The Wiener--Hopf Technique, its Generalisations and Applications: Constructive and Approximate Methods

“…In this section we will look at an infinite system of equations of Wiener-Hopf type, a discrete analogue of integral equation (1). For all infinite sequence a n considered here we will assume the following holds for some constant M…”

“…The study of this equation was principally motivated by an interest of one of the authors, in a differential equation governing the radiation equilibrium of stars [91]. The Wiener-Hopf equation arises from extending the integral equation (1)…”

“…Note that, although h(x) is unknown, it is not independent as it uniquely defined by the left hand side of (2) once f (x) is determined. Applying the Fourier transform to (2) results in an equation of type (3) (the interaction of a Fourier transform with its convolution has to be used, and analytic properties of half-range Fourier transforms employed 1 ). This laid the foundation for the study of scalar Wiener-Hopf equations.…”

“…The latter two authors also considered integral equations with kernels depending on the difference of arguments, but containing integrals with variable limits or along a finite interval, respectively. Nevertheless, the Wiener-Hopf integral equation (1) became an important independent subject of research due to its wide range of applicability and many connections. The Wiener-Hopf method has enabled researchers to analytically solve numerous previously intractable integral and partial differential equations, as well as many boundary value problems.…”

“…applied analysis together via the Wiener-Hopf technique, its generalisations and applications", where the most important results in the area were presented and further developments of the method were considered [1]. This review was inspired by that programme.…”