Gabor Analysis of the Spaces M(P,q,w) (Rd) and S(P,q,r,w,ω) (Rd)

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) (Rd) to be the subspace of tempered distributions f e{open} S'(Rd) such that the Gabor transform V<inf>g</inf>(f) of f is in the weighted Lorentz space L(p,q,wdμ)(R2d). 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≤∞ We also investigate the embeddings between these spaces and the dual space of M(p,q,w)(Rd). Later we define the space S(p,q,r,w,ω)Rd for 1 < p < ∞, 1 ≤ q ≤ ∞ 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,ω)(Rd). At the end of this article, we characterize the multipliers of the spaces M(p,q,w)(Rd) and S(p,q,r,w,ω)(Rd). © 2011 Wuhan Institute of Physics and Mathematics.