Harmonic Analysis: A Historical Manifold during the XXth Century

Abstract

Harmonic analysis is the modern version of classical Fourier analysis. By the end of the XlXth century it had become clear that convergence of a Fourier series is far from being automatic. The necessity of studying exceptional subsets was a main purpose for the introduction of abstract set theory.Hilbert’s fifth problem was about harmonic analysis, possible extensions of Lie’s theory on ‘continuous transformation groups’. Along the XXth century harmonic analysis was partaking of a variety of mathematical fields not directly linked to each other.In physics, quantum mechanics was put on solid mathematical grounds by von Neumann’s theory of representations of group actions. Before the power of abstract harmonic analysis had become available, Wiener launched his impressive work on Tauberian theorems leading to harmonic synthesis. The extension of Lebesgue’s theory to locally compact groups initiated by Haar eased the way to abstract harmonic analysis.Gelfand’s theory of Banach algebras was applied successfully to the study of group algebras. More generally, C*-algebras and von Neumann algebras became the framework for many problem in harmonic analysis.The recent theory of wavelets, Fourier-series like sums, provides a useful tool, a ‘microscope’, for scrutinizing localized phenomena of various natures.