In this paper modulation spaces as well as their relatives (generalized amalgam spaces, as discussed in the paper about "Generalized Amalgams") are described in terms of the behavior of the Short-Time Fourier Transform (or sliding window Fourier transform) of their elements. Among the members of this family of spaces are Bessel potential spaces derived from L^2, and the remarkable Segal algebra S_0(G).

It is the aim of this paper to derive so-called atomic decompositions of Gabor-type for the members of these function spaces, in the spirit of Frazier-Jawerth phi-transforms and the corresponding characterizations of Besov spaces. To be more precise, it is shown, that membership of a function in any of the spaces discussed can be completely be characterized by the possibility of expanding the functions in question into a (double) series, where the building blocks are just time-frequency shifted copies of some "mother atom", and the behavior of the (doubly infinite) coefficient scheme corresponds to the parameters of the space. Typically on has mixed-norm weighted lp-lq-conditions on the coefficients desribes the membership of a function/distributions in such a general modulation space.

In the paper use of amalgam spaces is made. Much more general results have been obtained later, cf. the contribution to the Chui-book "Gabor wavelets and the Heisenberg Group".  There is also a pre-runner concerning the general modulation spaces over LCA groups (Techn. Rep. 1983, this report is being retyped - late 2002 - and should be made available soon).

This article can be seen as a prerunner to a series of papers on general atomic decompositions (1988/89) by Feichtinger and Gröchenig, as well as a systematic further development of efficient methods for Gabor analysis (cf. the books by Feichtinger/Strohmer, 1998 and 2002,  and Gröchenig, 2001).

Keywords: function spaces, Gabor analysis, atomic decomposition, amalgam spaces,

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