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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Strictly positive definite functions on compact abelian groups


Authors: Jan Emonds and Hartmut Führ
Journal: Proc. Amer. Math. Soc. 139 (2011), 1105-1113
MSC (2010): Primary 43A25; Secondary 43A75
Published electronically: August 10, 2010
MathSciNet review: 2745662
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Abstract: We study the Fourier characterisation of strictly positive definite functions on compact abelian groups. Our main result settles the case $ G = F \times \mathbb{T}^r$, with $ r \in \mathbb{N}$ and where $ F$ is a finite abelian group. The characterisation obtained for these groups does not extend to arbitrary compact abelian groups; it fails in particular for all torsion-free groups.


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Additional Information

Jan Emonds
Affiliation: Institut für Mathematik, Universität Paderborn, D-33098 Paderborn, Germany
Email: emonds@math.upb.de

Hartmut Führ
Affiliation: Lehrstuhl A für Mathematik, RWTH Aachen, D-52056 Aachen, Germany
Email: fuehr@matha.rwth-aachen.de

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10533-6
PII: S 0002-9939(2010)10533-6
Keywords: Strictly positive definite functions, compact abelian groups; trigonometric polynomials, Fourier series
Received by editor(s): February 15, 2010
Received by editor(s) in revised form: April 9, 2010
Published electronically: August 10, 2010
Additional Notes: The first author was supported by the DFH and the International Research Training Group DFG-1133 “Geometry and Analysis of Symmetries”
Communicated by: Michael T. Lacey
Article copyright: © Copyright 2010 American Mathematical Society