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On random Hermite series


Authors: Rafik Imekraz, Didier Robert and Laurent Thomann
Journal: Trans. Amer. Math. Soc. 368 (2016), 2763-2792
MSC (2010): Primary 60G50
Published electronically: April 8, 2015
MathSciNet review: 3449257
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Abstract: We study integrability and continuity properties of random series of Hermite functions. We get optimal results which are analogues to classical results concerning Fourier series, like the Paley-Zygmund or the Salem-Zygmund theorems. We also consider the case of series of radial Hermite functions, which are not so well-behaved. In this context, we prove some$ L^p$ bounds of radial Hermite functions, which are optimal when $ p$ is large.


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Additional Information

Rafik Imekraz
Affiliation: Institut de Mathématiques de Bordeaux, UMR 5251 du CNRS, Université de Bordeaux 1, 351, cours de la Libération F33405 Talence Cedex, France
Email: rafik.imekraz@math.u-bordeaux1.fr

Didier Robert
Affiliation: Laboratoire de Mathématiques J. Leray, UMR 6629 du CNRS, Université de Nantes, 2, rue de la Houssinière, 44322 Nantes Cedex 03, France
Email: didier.robert@univ-nantes.fr

Laurent Thomann
Affiliation: Laboratoire de Mathématiques J. Leray, UMR 6629 du CNRS, Université de Nantes, 2, rue de la Houssinière, 44322 Nantes Cedex 03, France
Email: laurent.thomann@univ-nantes.fr

DOI: https://doi.org/10.1090/tran/6607
Keywords: Harmonic oscillator, Hermite functions, random series, harmonic analysis.
Received by editor(s): March 20, 2014
Published electronically: April 8, 2015
Additional Notes: The second author was partly supported by the grant “NOSEVOL” ANR-2011-BS01019 01
The third author was partly supported by the grant “HANDDY” ANR-10-JCJC 0109, and by the grant “ANAÉ” ANR-13-BS01-0010-03
Article copyright: © Copyright 2015 American Mathematical Society