Gabor analysis of the spaces M(p,q,w) (ℝ d) AND S(p,q,r,w,ω) (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 ɛ S'(ℝ d) such that the Gabor transform V g( f) of f is in the weighted Lorentz space L(p,q,wdμ)(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≤∞. 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,ω)ℝ d 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)(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,ω)(ℝ d).