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Describing functions: Atomic decompositions versus frames. (English) Zbl 0736.42022

The theory of frames and non-orthogonal series expansions with respect to coherent states is extended to a general class of spaces, the so-called coorbit spaces.

Let π be an integrable, irreducible, continuous representation of a locally compact group 𝒢 on a Hilbert space , g a suitable “test function”, and (x i ), iI, a sufficiently dense set in 𝒢. Then for the coorbit spaces series expansions are constructed, which are of the form f= i c i π(x i )g. The coefficients depend linearly and continuously on f. Conversely, f in a coorbit space is uniquely determined by the sampling of the representation coefficient π(x i )g,f and can be stably reconstructed from these values.

Special cases include wavelet expansions for the Besov-Triebel-Lizorkin spaces, Gabor-type expansions for modulation spaces and sampling theorems for wavelet and Gabor transforms, and series expansions and sampling theorems for certain spaces of analytic functions.

Reviewer: K.Gröchenig

MSC:
42C15General harmonic expansions, frames
43A15L p -spaces and other function spaces on groups, semigroups, etc.
46E99Linear function spaces and their duals
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