A theorem concerning Fourier transforms: A survey

TL;DR

This survey reviews the historical impact and subsequent developments of Hardy's 1933 theorem on Fourier transforms, highlighting its influence on harmonic analysis and related fields over the past 90 years.

Contribution

It provides a comprehensive overview of generalizations, connections, and new perspectives stemming from Hardy's theorem in harmonic analysis.

Findings

Multiple generalizations of Hardy's theorem have been developed.

The theorem has influenced various related areas in harmonic analysis.

New perspectives have emerged from the connections to other mathematical fields.

Abstract

In this note, we highlight the impact of the paper G. H. Hardy, A theorem concerning Fourier transforms, J. Lond. Math. Soc. (1) 8 (1933), 227--231 in the community of harmonic analysis in the last 90 years, reviewing, on the one hand, the direct generalizations of the main results and, on the other, the different connections to related areas and new perspectives.