Real Paley-Wiener theorems in spaces of ultradifferentiable functions

Abstract

We develop real Paley-Wiener theorems for classes S ω of ultradifferentiable functions and related L p -spaces in the spirit of Bang and Andersen for the Schwartz class. We introduce results of this type for the so-called Gabor transform and give a full characterization in terms of Fourier and Wigner transforms for several variables of a Paley-Wiener theorem in this general setting, which is new in the literature. We also analyze this type of results when the support of the function is not compact using polynomials. Some examples are given.2010 Mathematics Subject Classification. Primary 46F12, Secondary 46F05, 42B10.

Multipliers on $${{\mathcal {S}}}_{\omega }({{\mathbb {R}}}^N)$$

“…The space S ω (R N ) is a Fréchet space with different equivalent systems of seminorms. Indeed, the following result holds (see [4,Theorem 4.8] and [3,Theorems 2.6]).…”

“…In the last years the attention has focused on the study of the space S ω (R N ) of the ultradifferentiable rapidly decreasing functions of Beurling type, as introduced by Björck [2] (see, [3][4][5], for instance, and the references therein). Inspired by this line of research and by the previous work, in this paper we introduce and study the space O M,ω (R N ) of the slowly increasing functions of Beurling type in the setting of ultradifferentiable function space as introduced in [8].…”

“…In Sect. 3 we also introduce the space O C,ω (R N ) of the very slowly increasing functions of Beurling type. In particular, we show the link between these spaces and their topological properties.…”

Regularity of global solutions of partial differential equations in non isotropic ultradifferentiable spaces via time-frequency methods

“…At first, there is no loss of generality in assuming that ω| [0,1] ≡ 0; as a consequence, we easily have from (2.1) that ϕ * ω (0) = 0. Moreover the properties in the next proposition hold; they are well-known and can be found in many references, we refer for instance to [5], where (in Section 2 and in the Appendix) several basic properties of weights are collected and proved with minimal assumptions.…”

“…Moreover, the spaces S Σ Ω contain as particular cases Beurling spaces of Gelfand-Shilov type. We give different equivalent systems of seminorms for the space S Σ Ω (R N ), in the spirit of the results contained in [3,5], and we consider the problem of regularity in this setting. We say that an operator is S Σ Ω -regular if the conditions u ∈ (S Σ Ω ) ′ , Au ∈ S Σ Ω imply that u ∈ S Σ Ω .…”