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Gabor frames for \(L^ 2\) and related spaces. (English) Zbl 0887.42025

Benedetto, John J. (ed.) et al., Wavelets: mathematics and applications. Boca Raton, FL: CRC Press. Studies in Advanced Mathematics. 97-162 (1994).
The paper surveys the theory of Gabor-frames as a tool for developing atomic decomposition of Banach spaces. After reviewing the basic ideas of frames in Hilbert spaces, the Balian-Low theorem for Gabor frames is proved in the non-distributional case.
Further, the notion of a frame is exended to “Banach frames” in order to study atomic decomposition of Bessel potential spaces in terms of Gabor systems. The Zak transform is proved to be very useful in this context.
Finally, a Gabor decomposition of \(L^1(R)\) is represented. This decomposition is based on the continuous Gabor transform which can be seen as a generalization of the ordinary Fourier transform.
For the entire collection see [Zbl 0840.00013].
Reviewer: G.Plonka (Rostock)

MSC:

42C15 General harmonic expansions, frames
46C15 Characterizations of Hilbert spaces
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
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