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Extended Gevrey Regularity via the Short-Time Fourier Transform

  • Nenad TeofanovEmail author
  • Filip Tomić
Chapter
  • 54 Downloads
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Abstract

We study the regularity of smooth functions whose derivatives are dominated by sequences of the form \(M_p^{\tau ,\sigma }=p^{\tau p^{\sigma }}\), τ > 0, σ ≥ 1. We show that such functions can be characterized through the decay properties of their short-time Fourier transforms (STFT), and recover (Cordero et al., Trans. Am. Math. Soc., 367 (2015), 7639–7663; Theorem 3.1) as the special case when τ > 1 and σ = 1, i.e. when the Gevrey type regularity is considered. These estimates lead to a Paley-Wiener type theorem for extended Gevrey classes. In contrast to the related result from Pilipović et al. (Sarajevo Journal of Mathematics, 14 (2) (2018), 251–264; J. Pseudo-Differ. Oper. Appl. (2019)), here we relax the assumption on compact support of the observed functions. Moreover, we introduce the corresponding wave front set, recover it in terms of the STFT, and discuss local regularity in such context.

Keywords

Gevrey classes Paley-Wiener theorem Modulation spaces Wave front sets Ultradistributions 

Notes

Acknowledgements

This work is supported by MPNTR through Project 174024.

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Authors and Affiliations

  1. 1.Department of Mathematics and Informatics, Faculty of SciencesUniversity of Novi SadNovi SadSerbia
  2. 2.Department of Fundamental Sciences, Faculty of Technical SciencesUniversity of Novi SadNovi SadSerbia

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