Ultradistributions and Time-Frequency Analysis

Abstract

Abstract. The aim of the paper is to show the connection between the theory of ultradistributions and time-frequency analysis. This is done through timefrequency representations and modulation spaces. Furthermore, some classes of pseudo-differential operators are observed. Mathematics Subject Classification (2000). Primary 46F05, 47G30; Secondary 35S05, 44A05.

Super-exponential decay and holomorphic extensions for semilinear equations with polynomial coefficients

“…We also note that our results lead to generalizations for Gelfand-Shilov classes S μ ν (R n ) with 1/2 ν 1 of some regularity assertions for Shubin type pseudodifferential operators in timefrequency analysis (see Gröchenig, Zimmermann [15], Pilipovic, Teofanov [24], Teofanov [30] where ν is supposed to be greater than 1). Before passing to nonlinear equations, we shortly digress to semi-classical linear operators P (x, hD), see again [28].…”

Bilinear Localization Operators on Modulation Spaces

“…We refer to [20,21,[29][30][31] for the proof and more details on STFT in other spaces of Gelfand-Shilov type.…”

“…We restrict ourselves to ( , ) = ⟨ ⟩ ⟨ ⟩ , , ∈ R, since the convolution and multiplication estimates which will be used later on are formulated in terms of weighted spaces with such polynomial weights. As already mentioned, weights of exponential type growth are used in the study of GelfandShilov spaces and their duals in [16,[29][30][31]. We refer to [36] for a survey on the most important types of weights commonly used in time-frequency analysis.…”

Supports of Representations in the Cohen Class

“…to [11] or [19]. Particularly important is the invariance of the Gelfand-Shilov spaces S α α (R d ) under the Wigner transform (as well as under the STFT), more precisely (see [19] (Thm 3.8))…”