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 Aw B,Y(G) consisting of all complex-valued functions f ? L1 w,(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 Aw(G) LwP(G),Y is an abstract Segal algebra with respect to Lw 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 w(G) to Aw B,Y(G) is the space Aw B,Y(G). © 2000 Warsaw University. All rights reserved.