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Convolution and Anti-Wick Quantisation on Ultradistribution Spaces

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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 present recent advances in convolution theory for the quasi-analytic and non quasi-analytic ultradistribution spaces and generalised Gelfand-Shilov spaces. Additionally, we consider the existence of convolution of non-quasianalytic ultradistribution with the Gaussian kernel \(e^{s|x|{ }^2}\), \(s\in \mathbb R\backslash \{0\}\), and identify the largest subspace of non-quasi-analytic ultradistributions for which this convolution exists. This gives a way to extend the definition of Anti-Wick quantisation for symbols that are not necessarily tempered ultradistributions. Finally, we discuss convolution in quasi-analytic classes with \(e^{s|\cdot |{ }^q}\), q > 1, \(s\in \mathbb R\).

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Notes

  1. 1.

    here and throughout the rest of the article we apply the principle of vacuous products, i.e. \(\prod _{j=1}^0l_j=1\).

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Acknowledgements

The work of Stevan Pilipović was supported by the Ministry of Education and Science, Republic of Serbia, project no. 174024.

The work of B. Prangoski was partially supported by the bilateral project “Microlocal analysis and applications” funded by the Macedonian and Serbian academies of sciences and arts.

Author information

Authors and Affiliations

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

    Stevan Pilipović

  2. Department of Mathematics, Faculty of Mechanical Engineering-Skopje, Ss. Cyril and Methodius University in Skopje, Skopje, Macedonia

    Bojan Prangoski

Authors
  1. Stevan Pilipović
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  2. Bojan Prangoski
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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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Pilipović, S., Prangoski, B. (2020). Convolution and Anti-Wick Quantisation on Ultradistribution Spaces. 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_22

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