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

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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.

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Notes

  1. This definition of \(\mathbf{S }_{0}(G)\) leads to some immediate questions: Why are functions \(f\in L^{1}(G)\) that satisfy (1.1) continuous? Why is the set of functions satisfying (1.1) a subspace of \(L^{1}(G)\)? Why is the integral in the norm well-defined?

  2. If G is an elementary LCA group, i.e., \(G\cong \mathbb {R}^{l}\times \mathbb {Z}^{m}\times \mathbb {T}^{n}\times F\) for some finite Abelian group F and with \(l,m,n\in \mathbb {N}_{0}\), then the Schwartz–Bruhat space \(\mathcal {S}(G)\) is a Fréchet space. Otherwise, the Schwartz–Bruhat space is a locally convex inductive limit of Fréchet spaces, i.e., an LF-space.

  3. See, for example, [8, 9, 13, 25, 26, 28, 30, 33, 54, 65, 87, 89, 90, 94, 95, 98, 119, 148,149,150,151,152].

  4. See, for example, [52, 63, 64, 67, 69, 70, 74, 87, 91, 93, 96, 97, 101, 124].

  5. In the same publication Reiter calls \(\mathbf{S }_{0}\) the canonical Segal algebra.

  6. The quasimeasuresQ(G) are the dual space of \(C_{c}(G)\cap A(G)\) with its inductive limit topology [29, Sect. 3]. One can show that \(\mathbf{S }'_{0}(G) = \{ \sigma \in Q(G) : \vert \sigma (T_{x}g)\vert < \infty \}\) for any (all) \(g\in C_{c}(G)\cap A(G)\) [41]. This motivates the name translation bounded quasimeasures for \(\mathbf{S }'_{0}(G)\).

  7. The weak\(^{*}\) topology of the dual of a Banach space is metrizable only if the Banach space is finite dimensional. If \(\mathbf{S }_{0}(G)\) is separable (e.g., if G is \(\sigma \)-compact and metrizable) then the relative weak\(^{*}\) topology on bounded sets in \(\mathbf{S }'_{0}(G)\) is metrizable and one can work with sequences. Otherwise nets have to be considered. For more on this, we refer to the book by Megginson [126].

  8. If G is \(\sigma \)-compact and metrizable then the Haar measure on \(G\times \widehat{G}\) is \(\sigma \)-finite. Hence \((L^{1}(G\times \widehat{G}))'\) can be identified with \(L^{\infty }(G\times \widehat{G})\) in the usual way. If G is a general LCA group, then the Haar measure on \(G\times \widehat{G}\) may not be \(\sigma \)-finite. In order to still have the identification of \((L^{1})'\) with \(L^{\infty }\) one redefines \(L^{\infty }\) to be the set of all mesureable functions that are locally almost everywhere bounded. I.e., every function in \(L^{\infty }\) is bounded except on a set N, where for all compact sets K the intersection \(N\cap K\) has measure zero. See [75, Sect. 2.3] and [103, Sect. 12].

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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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    Mads S. Jakobsen

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Communicated by Karlheinz Gröchenig.

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Jakobsen, M.S. On a (No Longer) New Segal Algebra: A Review of the Feichtinger Algebra. J Fourier Anal Appl 24, 1579–1660 (2018). https://doi.org/10.1007/s00041-018-9596-4

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