Applebaum, D. (2017) Probabilistic trace and Poisson summation formulae on locally compact abelian groups. Forum Mathematicum, 29 (3). pp. 501-517. ISSN 0933-7741
Abstract
We investigate convolution semigroups of probability measures with continuous densities on locally compact abelian groups, which have a discrete subgroup such that the factor group is compact. Two interesting examples of the quotient structure are the d-dimensional torus, and the adèlic circle. Our main result is to show that the Poisson summation formula for the density can be interpreted as a probabilistic trace formula, linking values of the density on the factor group to the trace of the associated semigroup on L2-space. The Gaussian is a very important example. For rotationally invariant α-stable densities, the trace formula is valid, but we cannot verify the Poisson summation formula. To prepare to study semistable laws on the adèles, we first investigate these on the p-adics, where we show they have continuous densities which may be represented as series expansions. We use these laws to construct a convolution semigroup on the adèles whose densities fail to satisfy the probabilistic trace formula.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2016 Forum Mathematicum. This is an author produced version of a paper subsequently published in Forum Mathematicum. Uploaded in accordance with the publisher's self-archiving policy. |
Dates: |
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Institution: | The University of Sheffield |
Academic Units: | The University of Sheffield > Faculty of Science (Sheffield) > School of Mathematics and Statistics (Sheffield) |
Depositing User: | Symplectic Sheffield |
Date Deposited: | 14 Nov 2016 14:49 |
Last Modified: | 16 Mar 2020 15:48 |
Published Version: | https://doi.org/10.1515/forum-2016-0067 |
Status: | Published |
Publisher: | De Gruyter |
Refereed: | Yes |
Identification Number: | 10.1515/forum-2016-0067 |
Related URLs: | |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:106772 |
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Filename: ProbTrace 1.pdf
Filename: ProbTracecorrigendum1.pdf
Description: Corrigendum