Periodic distributions and periodic elements in modulation spaces

Abstract

We characterize periodic elements in Gevrey classes, Gelfand-Shilov distribution spaces and modulation spaces, in terms of estimates of involved Fourier coefficients, and by estimates of their short-time Fourier transforms. If q ∈ [1, ∞), ω is a suitable weight and (E E 0 ) ′ is the set of all E-periodic elements, then we prove that the dual of M ∞,q(

The Zak Transform on Gelfand–Shilov and Modulation Spaces with Applications to Operator Theory

“…We also let (E 0 E ) (R d ) be the set of all formal expansions in (1.17) and E 0 E (R d ) be the set of all formal expansions in (1.17) such that at most finite numbers of c( f , α) are non-zero (cf. [40]). We refer to [25,40] for more characterizations of E σ E , E σ ;0 E and their duals.…”

“…[40]). We refer to [25,40] for more characterizations of E σ E , E σ ;0 E and their duals. The following definition takes care of spaces of formal expansions (1.17) with coefficients obeying specific quasi-norm estimates.…”

“…for some constant c which depends on the sampling parameters. (See [21,40] and Sect. 1 for notations.)…”

Wiener Estimates on Modulation Spaces

“…In the paper we characterise Gelfand-Shilov spaces of functions and distributions, modulation spaces and Gevrey classes in background of various kinds of Wiener estimates. We apply the results to deduce some refined formulae on periodic functions and distributions, given in [24].…”

“…Here M ∞,q is the (unweighted) modulation spaces with Lebesgue parameters ∞ and q. (See Section 1 or [24] for notations.) We note that a proof of (0.1) in the case q ∈ [1, ∞] can be found in e. g. [21], and with some extensions in [19].…”