Gabor frames for L2 and related spaces

The basic theory of frames is reviewed, and special topics dealing with Gabor frames and decompositions are developed. These topics include Gabor decompositions of L1 and of Bessel potential spaces. (Sobolev spaces are Bessel potential spaces.) Frames of translates in L2 are characterized; and the Balian–Low theorem for L2 is proved. The former result is not only useful for the Gabor theory, but is the basis of multiresolution analysis frames; the latter result is related to the classical uncertainty principle inequality.

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