Wiener Amalgams Over Euclidean Spaces And Some of Their Applications

Wiener amalgams have been introduced by the author in 1980 ([F1,2]). 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 Land using global l-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 are explained, among them various results which use these spaces only for the proof, but not in their statements. Much of the material presented here was influenced by the experiences the author has made 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.Gröchenig).