## Banach spaces related to integrable group representations and their atomic decompositions. I.(English)Zbl 0691.46011

In recent years there has been substantial work done showing that functions in many Banach spaces of functions have “atomic” or “molecular” decompositions; that is, the function can be represented as a sum of coefficients times simple building blocks in such a way that the size of the coefficients captures a great deal of information about the function. In some cases such results are obtained by methods that emphasize the Fourier transform, in other cases the emphasis is function theoretic. In this paper the authors show that many such results can be unified and extended if the spaces considered are spaces on which an integrable group representation acts. The basic building blocks which are used can be taken from the orbit of a single function under the group action. Thus they are generalized coherent states.
The heart of the proof is the construction of an integral reproducing formula followed by discretizing the formula and giving appropriate norm estimates.
Later papers in the series will demonstrate how the decomposition theorems can be used to study the functional analysis of the spaces and will describe the relation of the abstract theory to classical examples.
Reviewer: R. Rochberg

### MSC:

 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 43A99 Abstract harmonic analysis 22D10 Unitary representations of locally compact groups 46F05 Topological linear spaces of test functions, distributions and ultradistributions
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