On Functions With Fourier Transforms in W(B,y)

Abstract

Let G be a locally compact abelian group, let G be the dual group G. Research on Wiener type spaces was initiated by N. Wiener in [11]. A number of authors worked on these spaces or some special cases of these spaces. A kind of generalization of the Wiener's definition was given by H. Feichtinger in [5], [7] its a Banach spaces of functions on locally compact groups that are defined by means of the global behavior of certain local properties of their elements. In this paper, the space A<inf>w</inf>B,Y(G) consisting of all complex-valued functions f ϵ L1<inf>w</inf>,(G) whose Fourier transforms f belong to the Wiener type spaces W(B,Y) is investigated, where w is Beurling weights on G (c.f. [9]). In the first section invariance properties and asymptotic estimates for the translation and modulation operators are given. Furthermore the closed ideals are discussed and it is showed that the space A<inf>w(G)</inf>LwP(G),Y is an abstract Segal algebra with respect to L<inf>w</inf>1(G). At the end of this work, it is proved that if G is a locally compact abelian group then the space of all multipliers from L1<inf>w</inf>(G) to A<inf>w</inf>B,Y(G) is the space A<inf>w</inf>B,Y(G). © 2000 Warsaw University. All rights reserved.