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Moment Functions on Affine Groups

  • Żywilla FechnerEmail author
  • László Székelyhidi
Open Access
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Abstract

Moments of probability measures on a hypergroup can be obtained from so-called (generalized) moment functions of a given order. The aim of this paper is to characterize generalized moment functions on a non-commutative affine group. We consider a locally compact group G and its compact subgroup K. First we recall the notion of the double coset space G /  / K of a locally compact group G and introduce a hypergroup structure on it. We present the connection between K-spherical functions on G and exponentials on the double coset hypergroup G /  / K. The definition of the generalized moment functions and their connection to the spherical functions is discussed. We study an important class of double coset hypergroups: specyfing K as a compact subgroup of the group of inverible linear transformations on a finitely dimensional linear space V we consider the affine group \({\mathrm {Aff\,}}K\). Using the fact that in the finitely dimensional case \(({\mathrm {Aff\,}}K,K)\) is a Gelfand pair we give a description of exponentials on the double coset hypergroup \({\mathrm {Aff\,}}K//K\) in terms of K-spherical functions. Moreover, we give a general description of generalized moment functions on \({\mathrm {Aff\,}}K\) and specific examples for \(K=SO(n)\), and on the so-called \(ax+b\)-group.

Keywords

Hypergroup generalized moment function affine group spherical functions 

Mathematics Subject Classification

20N20 43A62 39B99 

Notes

Acknowledgements

The study was funded by Hungarian National Foundation for Scientific Research (OTKA) with Grant No. K111651.

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Copyright information

© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Institute of MathematicsLodz University of TechnologyŁódźPoland
  2. 2.University of DebrecenDebrecenHungary

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