Wiener Amalgams over Euclidean Spaces and Some of Their Applications
Hans G. Feichtinger
Abstract
Wiener amalgams were introduced by the author in 1980 ([Fl,2]). The concept was aimed at the possibility of describing local and global properties of a function or distribution independently (allowing to speak of increasing smoothness at infinity ...). For a very readable survey of Ordinary’ amalgams, built up on local Lp - and using global f-spaces, see the article by J. J.Fournier and J.Stewart [FSt], explaining applications of amalgams to a wide range of problems in analysis. In the present paper further applications of Wiener amalgam spaces will be explained, among them results (such as Thm.8 or several results in [F7]) which use these spaces only for the proof, but not in their statements. We also explain the properties of two families of operators, used for spline approximation and discretization of measures in the context of function spaces. Much of the material presented here was influenced by the author’s experience in using Wiener amalgam spaces while proving results on atomic decompositions and during the work on the irregular sampling problem of functions (joint papers with K. Gròchenig). Besides the new results presented in this paper the reader will find a number of applications of Wiener amalgam spaces to questions arising in signal analysis, especially in regular and irregular sampling theory, in the paper [F7] in this volume.