From Sommerfeld Diffraction Problems to Operator Factorisation

Abstract

This article presents a brief survey devoted to the interplay between diffraction and operator theory, particularly between the so-called canonical diffraction problems (exemplified by Sommerfeld half-plane problems) on one hand and operator factorisation theory on the other hand. It is shown how operator factorisation concepts appear in a natural way from applications and how they can help to find solutions rigorously in case of well-posed problems or how to normalise problems by an adequate change of function spaces.

Reproducing Kernel Hilbert Space Associated with a Unitary Representation of a Groupoid

“…Another interesting problem might be to describe the so-called (unitary) equivalence after extension between two bounded linear operators in Hilbert spaces in terms of groupoids and their representations (see [24]). This may be applicable in the theory of partial differential equations and will be investigated in future work.…”

The Wiener–Hopf technique, its generalizations and applications: constructive and approximate methods

“…In the case of a classical matrix Wiener–Hopf equations the first one is straightforward while the remaining points continue to offer future avenues of research. It is possible to develop alternative methods for solving equations of Wiener–Hopf-type, for example based on Fredholm integral equation theory [140,175,176], and also more theoretical operator based approaches to convolution integral equations [154,177].…”

“…For this question, we have to specify in what sense are Kfalse(αfalse) and Kfalse¯false(αfalse) close. It is not an easy task to decide what the correct norm is to consider, but good candidates are Lp norms or Sobolev spaces [154]. We will review some known facts about approximation by rational functions, mostly by Padé approximants.…”

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

“…In the case of a classical matrix Wiener-Hopf equations the first one is straightforward while the remaining points continue to offer future avenues of research. It is possible to develop alternative methods for solving equations of Wiener-Hopf type, for example based on Fredholm integral equation theory [54,59,60], and also more theoretical operator based approaches to convolution integral equations [155,168].…”

“…For this question we have to specify in what sense are K(α) and K(α) close. It is not an easy task to decide what the correct norm is to consider, but good candidates are L p norms or Sobolev spaces [168]. We will review some known facts about approximation by rational functions, mostly by Padé approximants.…”