International Journal of Mathematics and Mathematical Sciences
Volume 13 (1990), Issue 3, Pages 517-525
doi:10.1155/S0161171290000758
On a family of weighted convolution algebras
1Institut für Mathematik, Universiät Wien, Strudlhofgasse 4, Wien A-1090, Austria
2Ondokuz Mayis University, Faculty of Art and Sciences, Department of Mathematics, Samsun, Turkey
Received 7 March 1988
Copyright © 1990 Hans G. Feichtinger and A. Turan Gürkanli. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Continuing a line of research initiated by Larsen, Liu and Wang [12], Martin and Yap [13], Gürkanli [15], and influenced by Reiter's presentation of Beurling and Segal algebras in Reiter [2,10] this paper presents the study of a family of Banach ideals of Beurling algebras Lw1(G), G a locally compact Abelian group. These spaces are defined by weighted Lp-conditions of their Fourier transforms. In the first section invariance properties and asymptotic estimates for the translation and modulation operators are given. Using these it is possible to characterize inclusions in section 3 and to show that two spaces of this type coincide if and only if their parameters are equal. In section 4 the existence of approximate identities in these algebras is established, from which, among other consequences, the bijection between the closed ideals of these algebras and those of the corresponding Beurling algebra is derived.