Time Frequency Analysis and Multipliers of the Spaces M(P, Q)(R D) and S(P, Q)(Rd)

Abstract

In the second section of this paper, in analogy to modulation spaces, we define the space M(p, q) (Rd) to be the subspace of tempered distributions f ∈ S′ (Rd) such that the Gabor transform V<inf>g</inf> (f) of f is in the Lorentz space L (p, q) (R2d), where the window function g is a rapidly decreasing function. We endow this space with a suitable norm and show that the M(p, q) (Rd) becomes a Banach space and is invariant under time-frequency shifts for 1 ≤ p, q ≤ ∞. We also discuss the dual space of M(p, q) (Rd) and the multipliers from L1 (Rd) into M(p, q) (Rd). In the third section we intend to study the intersection space S (p, q) (Rd) = L1 (Rd) ∩ M (p, q) (Rd) for 1 < p < ∞, 1 ≤ q ≤ ∞. We endow it with the sum norm and show that S (p, q) (Rd) becomes a Banach convolution algebra. Further we prove that it is also a Segal algebra. In the last section we discuss the multipliers of S (p, q) (Rd) and M (p, q) (Rd).