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Random Tempered Distributions on Locally Compact Separable Abelian Groups

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Operator Theory and Harmonic Analysis (OTHA 2020)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 358))

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Abstract

By means of sequences of random variables of controlled growth, that is, either rapidly decreasing or slowly increasing, we define a space of random tempered generalized functions—of Schwartz type—on a locally compact abelian separable group, and we characterize the set of those generalized functions that have a mean. We show that this approach covers both the case of the torus and the real line case and also allows us to define tempered random distributions over the Cantor set by means of the associated Cantor group.

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Acknowledgements

This work was partially supported, for the first author, through the project of the Centro de Matemática e Aplicações, UID/MAT/00297/2020, financed by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) and, for the second author, through Russian Foundation for Basic Research (RFBR) (grant no. 19-01-00451).

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Authors and Affiliations

  1. FCT Nova & CMA, UNL, Caparica, Portugal

    Manuel L. Esquível

  2. Don State Technical University, Rostov-on-Don, Russian Federation

    Nadezhda P. Krasii

Authors
  1. Manuel L. Esquível
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  2. Nadezhda P. Krasii
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Correspondence to Manuel L. Esquível .

Editor information

Editors and Affiliations

  1. Institute of Mathematics, Mechanic and Computer Sciences and Regional Mathematical Center, Southern Federal University, Rostov-on-Don, Russia

    Alexey N. Karapetyants

  2. Department of Higher Mathematics, Don State Technical University, Rostov-on-Don, Russia

    Igor V. Pavlov

  3. Steklov Mathematical Institute, Russian Academy of Sciences, Moscow, Russia

    Albert N. Shiryaev

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Esquível, M.L., Krasii, N.P. (2021). Random Tempered Distributions on Locally Compact Separable Abelian Groups. In: Karapetyants, A.N., Pavlov, I.V., Shiryaev, A.N. (eds) Operator Theory and Harmonic Analysis. OTHA 2020. Springer Proceedings in Mathematics & Statistics, vol 358. Springer, Cham. https://doi.org/10.1007/978-3-030-76829-4_7

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