# The Feichtinger Algebra

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## Abstract

We are now going to address a first topic from the theory of modulation spaces, which was initiated by Feichtinger in the early 1980’s: the *Feichtinger algebra S* _{0}(ℝ^{ n }) (we will study the general notion of modulation space in the next chapter). The elements of *S* _{0}(ℝ^{ n }) are characterized by the property that their Wigner transform is in *L* ^{1}(ℝ^{ n ⊕ } ℝ^{ n }), but it is not obvious at all that, with *this* definition, *S* _{0}(ℝ^{ n }) is a vector space, even less an algebra! We will therefore need an alternative, more tractable, equivalent definition.

## Keywords

Banach Algebra Inversion Formula Modulation Space Gabor Frame Weyl Operator
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© Springer Basel AG 2011