Gabor Analysis for a Broad Class of Quasi-Banach Modulation Spaces

Abstract

Abstract. We extend the Gabor analysis in [13] to a broad class of modulation spaces, allowing more general mixed quasi-norm estimates and weights in the definition of the modulation space quasi-norm. For such spaces we deduce invariance and embedding properties, and that the elements admit reconstructible sequence space representations using Gabor frames.

Periodic distributions and periodic elements in modulation spaces

“…(Cf. e. g. [7,9,14,27].) For example, by [9,27] it follows that if B in Definition 1.7 is a mixed quasi-normed space of Lebesgue type and φ ∈ M r…”

Liftings for ultra-modulation spaces, and one-parameter groups of Gevrey-type pseudo-differential operators

“…(ω) and M p = M p (ω) . The following proposition is a consequence of well-known facts in [12,20,21,54].…”

Wiener Estimates on Modulation Spaces

“…(Cf. e. g. [4,9,12,22].) For example, let p, q, E, ω and v be the same as in Definition 1.9 and 1.10, and let B = L p,q E (R 2d ) and r = min (1, p, q).…”

“…The next result is a reformulation of [22,Proposition 3.4], and indicates how Wiener spaces are connected to modulation spaces. The proof is therefore omitted.…”

“…We manage our more general situation by using techniques based on ideas in [9,17,18,22] and which can handle Lebesgue and Wiener spaces which are quasi-Banach spaces but may fail to be Banach spaces. Especially we shall follow a main idea in [9,22] and replace the usual convolution, used in [3,5,6,12], by a semi-continuous version which is less sensitive when convexity is lacking in the topological structures. For the semi-continuous convolution we deduce in Section 2 the needed Lebesgue and Wiener estimates.…”