Moment Functions on Affine Groups

  • Żywilla FechnerEmail author
  • László Székelyhidi
Open Access


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.


Hypergroup generalized moment function affine group spherical functions 

Mathematics Subject Classification

20N20 43A62 39B99 



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


  1. 1.
    Aczél, J.: Functions of binomial type mapping groupoids into rings. Math. Z. 154, 115–124 (1977)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Artzy, R.: Linear Geometry. Addison-Wesley, Reading (1965)zbMATHGoogle Scholar
  3. 3.
    Bloom, W.R., Heyer, H.: Harmonic Analysis of Probability Measures on Hypergroups. de Gruyter Studies in Mathematics, vol. 20. Walter de Gruyter & Co., Berlin (1995)CrossRefGoogle Scholar
  4. 4.
    Dieudonné, J.: Treatise on Analysis. Vol. VI, Translated from the French by I. G. Macdonald. Pure and Applied Mathematics. Academic Press, New York, London (1978)zbMATHGoogle Scholar
  5. 5.
    Fechner, Ż., Székelyhidi, L.: Sine functions on hypergroups. Arch. Math. (Basel) 106(4), 371–382 (2016)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Fechner, Ż., Székelyhidi, L.: Functional equations on double coset hypergroups. Ann. Funct. Anal. 8(3), 411–423 (2017)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Fechner, Ż., Székelyhidi, L.: Sine and cosine equations on hypergroups. Banach J. Math. Anal. 11(4), 808–824 (2017)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Hermann, P.: Representations of double coset hypergroups and induced representations. Manuscr. Math. 88(1), 1–24 (1955)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Hewitt, E., Ross, K.A.: Abstract Harmonic Analysis. Vol. I. Fundamental Principles of Mathematical Sciences, vol. 115. Springer, Berlin (1979)CrossRefGoogle Scholar
  10. 10.
    Lyndon, R.C.: Groups and Geometry. London Mathematical Society Lecture Note Series, vol. 101. Cambridge University Press, Cambridge (1985)CrossRefGoogle Scholar
  11. 11.
    Orosz, Á., Székelyhidi, L.: Moment functions on polynomial hypergroups in several variables. Publ. Math. Debr. 65(3–4), 429–438 (2004)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Orosz, Á., Székelyhidi, L.: Moment functions on polynomial hypergroups. Arch. Math. 85(2), 141–150 (2005)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Orosz, Á., Székelyhidi, L.: Moment functions on Sturm–Liouville hypergroups. Ann. Univ. Sci. Bp. Sect. Comput. 29, 141–156 (2008)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Székelyhidi, L.: Functional Equations on Hypergroups. World Scientific Publishing Co. Pte. Ltd., Hackensack, London (2012)zbMATHGoogle Scholar
  15. 15.
    Székelyhidi, L.: On spectral synthesis in several variables. Adv. Oper. Theory 2(2), 179–191 (2017)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Székelyhidi, L.: Spherical spectral synthesis. Acta Math. Hung. 153(1), 120–142 (2017)MathSciNetCrossRefGoogle Scholar
  17. 17.
    van Dijk, G.: Introduction to Harmonic Analysis and Generalized Gelfand Pairs. de Gruyter Studies in Mathematics. Walter de Gruyter & Co., Berlin (2009)CrossRefGoogle Scholar
  18. 18.
    Wolf, J.: Spherical functions on Euclidean space. J. Funct. Anal. 239, 127–136 (2006)MathSciNetCrossRefGoogle Scholar

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© The Author(s) 2018

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Authors and Affiliations

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

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