GABOR ANALYSIS OF THE SPACES M (p, q, w) (R-d) AND S (p, q, r, w, omega) (R-d)

Abstract

Let g be a non-zero rapidly decreasing function and w be a weight function. In this article in analog to modulation space, we define the space M(p, q, w)(R-d) to be the subspace of tempered distributions f E S'(R-d) such that the Gabor transform V-g(f) of f is in the weighted Lorentz space L(p, q, wd mu) (R-2d). We endow this space with a suitable norm and show that it becomes a Banach space and invariant under time frequence shifts for 1 <= p, q < infinity. We also investigate the embeddings between these spaces and the dual space of M(p, q, w)(R-d). Later we define the space S(p, q, r, w, omega)(R-d) for 1 < p < infinity, 1 <= q <= infinity. We endow it with a sum norm and show that it becomes a Banach convolution algebra. We also discuss some properties of S(p, q, r, w, omega)(R-d). At the end of this article, we characterize the multipliers of the spaces M(p,q, w)(R-d) and S(p,q,r,w,omega)(R-d).