Banach spaces related to integrable group representations and their atomic decompositions. I
type: Item
instance of: scholarly article
title: Banach spaces related to integrable group representations and their atomic decompositions. I
zbMATH Open document ID: 0691.46011
DOI: 10.1016/0022-1236(89)90055-4
author: H. G. Feichtinger
author: Karlheinz Gröchenig
published in: Journal of Functional Analysis
publication date: 1989-01-01T00:00:00Z
review text: 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.
reviewed by: Richard Rochberg
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 43A99
Mathematics Subject Classification ID: 22D10
Mathematics Subject Classification ID: 46F05
zbMATH DE Number: 4131015
zbMATH Keywords: ``atomic'' or ``molecular'' decompositions
zbMATH Keywords: integrable group representation
zbMATH Keywords: orbit
zbMATH Keywords: integral reproducing formula
zbMATH Keywords: decomposition theorems
MaRDI profile type: Publication
full work available at URL: 0022-1236(89)90055-4
OpenAlex ID: W1998630480
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description: scientific article; zbMATH DE number 4131015