Vol. 100, No. 2, 1982

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Functions of translation type and solid Banach spaces of functions

Reinhard Bürger

Vol. 100 (1982), No. 2, 259–269

Functions of translation type were introduced by H. Reiter and studied by the author in some detail. In this paper we introduce a class of Banach spaces of functions on a locally compact group, including the spaces p(G) of Liu-van Rooij-Wang. Necessary and sufficient conditions are given under which these spaces can be characterized by functions of translation type. As application it is shown that a sub-class of these spaces, including Wiener’s algebra, satisfies a certain minimality property. Furthermore, we obtain a generalization of a theorem on Fourier transforms, due to Edwards-Hewitt-Ritter, in a very simple manner, whereas the original proof took several pages.

Mathematical Subject Classification 2000
Primary: 43A15
Secondary: 46E99
Received: 5 August 1980
Revised: 26 March 1981
Published: 1 June 1982
Reinhard Bürger