Orthogonal Fourier Analysis on Domains

TL;DR

This paper reviews the historical development of the Fuglede Spectral Set Conjecture over the past 30 years, highlighting key developments, open questions, and areas not covered, with a focus on spectral sets and measures.

Contribution

It provides a personal, descriptive overview of the spectral set conjecture's history, emphasizing open questions and areas of ongoing research.

Findings

Historical overview of the Spectral Set Conjecture

Identification of open questions in the field

Discussion of developments in spectral measures

Abstract

In this paper we go over the history of the Fuglede or Spectral Set Conjecture as it has developed over the last 30 years or so. We do not aim to be exhaustive and we do not cover important areas of development such as the results on the problem in classes of finite groups or the version of the problem that focuses on spectral measures instead of sets. The selection of the material has been strongly influenced by personal taste, history and capabilities. We are trying to be more descriptive than detailed and we point out several open questions.