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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 \(\pi\) be an integrable, irreducible, continuous representation of a locally compact group \({\mathcal G}\) on a Hilbert space \({\mathcal H}\), \(g\in{\mathcal H}\) a suitable “test function”, and \((x_ i)\), \(i\in I\), a sufficiently dense set in \({\mathcal G}\). Then for the coorbit spaces series expansions are constructed, which are of the form \(f=\sum_ ic_ i\pi(x_ i)g\). The coefficients depend linearly and continuously on \(f\). Conversely, \(f\) in a coorbit space is uniquely determined by the sampling of the representation coefficient \(\langle\pi(x_ i)g,f\rangle\) 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:

42C15 General harmonic expansions, frames
43A15 \(L^p\)-spaces and other function spaces on groups, semigroups, etc.
46E99 Linear function spaces and their duals
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