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Extended Gevrey Regularity via the Short-Time Fourier Transform

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Advances in Microlocal and Time-Frequency Analysis

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

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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.

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Acknowledgements

This work is supported by MPNTR through Project 174024.

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Authors and Affiliations

  1. Department of Mathematics and Informatics, Faculty of Sciences, University of Novi Sad, Novi Sad, Serbia

    Nenad Teofanov

  2. Department of Fundamental Sciences, Faculty of Technical Sciences, University of Novi Sad, Novi Sad, Serbia

    Filip Tomić

Authors
  1. Nenad Teofanov
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  2. Filip Tomić
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Correspondence to Nenad Teofanov .

Editor information

Editors and Affiliations

  1. Dipartimento di Matematica “G. Peano”, Università degli Studi di Torino, Torino, Italy

    Paolo Boggiatto

  2. Dipartimento di Matematica “G. Peano”, Università degli Studi di Torino, Torino, Italy

    Marco Cappiello

  3. Dipartimento di Matematica “G. Peano”, Università degli Studi di Torino, Torino, Italy

    Elena Cordero

  4. Dipartimento di Matematica “G. Peano”, Università degli Studi di Torino, Torino, Italy

    Sandro Coriasco

  5. Dipartimento di Matematica “G. Peano”, Università degli Studi di Torino, Torino, Italy

    Gianluca Garello

  6. Dipartimento di Matematica “G. Peano”, Università degli Studi di Torino, Torino, Italy

    Alessandro Oliaro

  7. Dipartimento di Matematica "G. Peano", Università degli Studi di Torino, Torino, Italy

    Jörg Seiler

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Teofanov, N., Tomić, F. (2020). Extended Gevrey Regularity via the Short-Time Fourier Transform. In: Boggiatto, P., et al. Advances in Microlocal and Time-Frequency Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-36138-9_25

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