Feichtinger, Hans G.;

Modulation spaces of locally compact Abelian groups

(Radha, R.;Krishna, M.;Thangavelu, S. ed.) in Proc. Internat. Conf. on Wavelets and Applications New Delhi Allied Publishers (2003) [Chennai, January 2002] p.1-56

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abstract

This HTML-file has been formulated in Sept. 2002, after a number of requests from different colleagues revealed increased interest in modulations spaces and copies of the report. Meanwhile modulation spaces have become a \'standard\' tool in time-frequency analysis and many authors are using those spaces. See Charly\'s book

Karlheinz Gröchenig: Foundations of Time-Frequency Analysis. Birkhäuser, 2002, ISBN 0-8176-4022-3 (see cite{gr01})
for details on modulation spaces and time-frequency analysis (from a functional analytic point of vew in general)


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This 52pg. report summarizes the basic facts concerning modulation spaces in \'full generality\' (i.e. not only for polynomial weights, but rather for general weight functions, including the sub-exponential ones, satisfying only the so-called BD = Beurling-Domar non-quasianalyticity condition). Since the corresponding theory is naturally explained in the context of LCA groups this general frame-work has been choosen (not so much for the sake of abstract generality).

The modulation spaces in this context are defined in analogy with Besov spaces, using BUPUs, i.e. bounded uniform partitions of unity (consisting perferably of a sufficient smooth function with compact support, which together with its translates makes up a uniform partition of unity). In other words the modulation spaces derived from L^p have a description (on the FT side) as Wiener amalgam spaces (see cite{fe81-1}) with a local component of the form FL^p. Of course one has to make sure that for modulation spaces with such general weights (among them also spaces which are nowadays called \'ultra-modulation\' spaces, discussed in great detail in recent work of Nenad Teofanov ) there is a well defined Fourier transform (in the sense of ultra-distributions). The corresponding minimal modulation spaces (in the Gabor context their sequence space is a weighted l^1-space with separable weights) has to be discussed first. in \'Can.J.Math.\', and serving as a modul within the general coorbit theory (see cite{fegr89}).

It is shown that these spaces are well defined (independently from the BUPU which is used), that they can be characterized in a continuous way (nowadays interpreted as characterization of those spaces using the continuous STFT). The report also contains a characterization of compact sets in modulation spaces, trace theorems (similar to the Besov spaces), a result which has not been published elsewhere so far (compare however the papers by H.Triebel which appeared about at the same time, in the Euclidean setting, but also for p < 1).

There is also an imbedding theorem corresponding to Sobolev\'s embedding theorem.

Maybe a more detailed report will be formulated later on.

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