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Zbl 0736.42022
Gröchenig, Karlheinz
Describing functions: Atomic decompositions versus frames.
(English)
[J] Monatsh. Math. 112, No.1, 1-42 (1991). ISSN 0026-9255; ISSN 1436-5081/e

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.\par Let $\pi$ be an integrable, irreducible, continuous representation of a locally compact group ${\cal G}$ on a Hilbert space ${\cal H}$, $g\in{\cal H}$ a suitable test function'', and $(x\sb i)$, $i\in I$, a sufficiently dense set in ${\cal G}$. Then for the coorbit spaces series expansions are constructed, which are of the form $f=\sum\sb ic\sb i\pi(x\sb i)g$. The coefficients depend linearly and continuously on $f$. Conversely, $f$ in a coorbit space is uniquely determined by the sampling of the representation coefficient $\langle\pi(x\sb i)g,f\rangle$ and can be stably reconstructed from these values.\par 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.
[K.Gröchenig]
MSC 2000:
*42C15 Series and expansions in general function systems
43A15 Lp-spaces and other function spaces on groups, etc.
46E99 Linear function spaces and their duals

Keywords: atomic decompositions; integrable group representations; frames; coherent states; coorbit spaces; wavelet expansions; Besov-Triebel-Lizorkin spaces; Gabor-type expansions; modulation spaces; sampling theorems; Gabor transforms

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