Feichtinger, Hans G.
Banach convolution algebras of Wiener type
in Proc. Conf. on Functions, Series, Operators, Budapest 1980 North-Holland Colloq. Math. Soc. Janos Bolyai Vol.35 (1983) [Amsterdam] p.509--524, Zbl:0528.43001, MR0751019abstract
This is the basic paper containing the (first) definition of general Wiener amalgam space (at that time called Wiener-type spaces). These spaces are playing an important role within the subsequent work by the author:A) For the development of the general theory of atomic decompositions with respect to integrable group representations (with K. Gröchenig), described in a series of papers, of which \\'{ } Banach spaces related to integrable group representations and their atomic decompositions,I.\\'{ } (see cite{fegr89}) is the most important one.
B) Amalgam spaces also play a crucial role for the (related) Feichtinger-Gröchenig approach to the irregular sampling problem for band-limited functions.
In this first paper on Wiener amalgam spaces the equivalence of \\'{ }continuous\\'{ } and \\'{ }discrete\\'{ } description (making use of BUPUs = bounded admissible partitions of unity) for space W(B,C), on general locally compact groups, with general local component B and global component C is given. Also, the important convolution theorems for Wiener amalgam spaces are derived, based on the concept of a control function with respect to some window (generating the localization).
Further results on interpolation and duality are given in subsequent papers, such as \\'{ }Banach spaces of distributions of Wiener\\'{ }s type and interpolation\\'{ } (see cite{fe81-1}) and the joint paper with P.Gröbner on decomposition spaces (see cite{fegr85}). A good summary of Wiener amalgam spaces and their basic properties is the master thesis of Thomas Dobler, Vienna, Summer 1989 (see cite{do89}).
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