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Journal of Fourier Analysis and Applications

, Volume 24, Issue 6, pp 1579–1660 | Cite as

On a (No Longer) New Segal Algebra: A Review of the Feichtinger Algebra

  • Mads S. Jakobsen
Article
  • 158 Downloads

Abstract

Since its invention in 1979 the Feichtinger algebra has become a 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 develops the theory of Feichtinger’s algebra in a linear and comprehensive way. The material gives an entry point into the subject and it will also bring new insight to the expert. A further 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 this space, over operators on it, and to aspects of its dual space. Additional results include a seemingly forgotten theorem by Reiter on Banach space isomorphisms of the Feichtinger algebra, a new identification of Feichtinger’s 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 a list of related function space constructions is included.

Keywords

Feichtinger algebra Test functions Generalized functions Schwartz–Bruhat space 

Mathematics Subject Classification

Primary 43A15 Secondary 43-02 

Notes

Acknowledgements

The author thanks Hans G. Feichtinger for many invaluable discussions on his algebra and related topics. Furthermore, the author thanks Ole Christensen for support with the writing and the presentation of the material. Thanks also goes to Franz Luef, Jakob Lemvig, Jordy T. van Velthoven and the referees for numerous helpful comments. The author gratefully acknowledges that this work was partially carried out during the tenure of an ERCIM ‘Alain Bensoussan’ Fellowship Programme at NTNU.

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Authors and Affiliations

  1. 1.NTNU, Department of Mathematical SciencesTrondheimNorway

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