GABOR MULTIPLIERS FOR BANACH SPACES OF DISTRIBUTIONS ON LOCALLY COMPACT ABELIAN GROUPS

We define a weighted Banach space of test functions and its antidual H on a locally compact abelian group , so as to from a Gelfand triple . We obtain atomic characterization of in terms of Gabor atoms and study the boundedness properties of Gabor multipliers between these function spaces. Our Theorem 5.1 includes the corresponding results of Feichtinger and Zimmermann [FZ 98,p.l44] as particular cases. Also, we study the compactness property of Gabor multipliers on the space and , 1 ≤ p < ∞, and demonstrate that the space of all sequences of bounded variation is continuously embedded in the space of all Gabor multipliers on . This paper paves the way for the study of a number of other properties of Gabor multipliers defined on various types of Banach spaces of functions and distributions.