On a (no longer) new Segal algebra - a review of the Feichtinger algebra

TL;DR

This review comprehensively discusses the Feichtinger algebra, highlighting its properties, multiple characterizations, and applications in time-frequency analysis and operator theory, providing both entry-level insights and new theoretical results.

Contribution

The paper offers a unified presentation of the Feichtinger algebra's various characterizations, introduces new identifications, and revisits classical theorems with modern insights.

Findings

Equivalence of different characterizations of the Feichtinger algebra

A new identification of the algebra as a unique Banach space in L^1

A kernel theorem for the Feichtinger algebra

Abstract

Since its invention in 1979, the Feichtinger algebra has become a very useful Banach space of functions with applications in time-frequency analysis, the theory of pseudo-differential operators and several other topics. It is easily defined on locally compact abelian groups and, in comparison with the Schwartz(-Bruhat) space, the Feichtinger algebra allows for more general results with easier proofs. This review paper gives a linear and comprehensive deduction of the theory of the Feichtinger algebra and its favourable properties. The material gives an entry point into the subject, but it will also give new insight to the expert. A main goal of this paper is to show the equivalence of the many different characterizations of the Feichtinger algebra known in the literature. This task naturally guides the paper through basic properties of functions that belong to the space, over operators on it and to aspects of its dual space. Further results include a seemingly forgotten theorem by Reiter on operators which yield Banach space isomorphisms of the Feichtinger algebra; a new identification of the Feichtinger algebra as the unique Banach space in $L^{1}$ with certain properties; and the kernel theorem for the Feichtinger algebra. A historical description of the development of the theory, its applications and related function space constructions is included.