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.
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References
Aczél, J.: Functions of binomial type mapping groupoids into rings. Math. Z. 154, 115–124 (1977)
Artzy, R.: Linear Geometry. Addison-Wesley, Reading (1965)
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)
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)
Fechner, Ż., Székelyhidi, L.: Sine functions on hypergroups. Arch. Math. (Basel) 106(4), 371–382 (2016)
Fechner, Ż., Székelyhidi, L.: Functional equations on double coset hypergroups. Ann. Funct. Anal. 8(3), 411–423 (2017)
Fechner, Ż., Székelyhidi, L.: Sine and cosine equations on hypergroups. Banach J. Math. Anal. 11(4), 808–824 (2017)
Hermann, P.: Representations of double coset hypergroups and induced representations. Manuscr. Math. 88(1), 1–24 (1955)
Hewitt, E., Ross, K.A.: Abstract Harmonic Analysis. Vol. I. Fundamental Principles of Mathematical Sciences, vol. 115. Springer, Berlin (1979)
Lyndon, R.C.: Groups and Geometry. London Mathematical Society Lecture Note Series, vol. 101. Cambridge University Press, Cambridge (1985)
Orosz, Á., Székelyhidi, L.: Moment functions on polynomial hypergroups in several variables. Publ. Math. Debr. 65(3–4), 429–438 (2004)
Orosz, Á., Székelyhidi, L.: Moment functions on polynomial hypergroups. Arch. Math. 85(2), 141–150 (2005)
Orosz, Á., Székelyhidi, L.: Moment functions on Sturm–Liouville hypergroups. Ann. Univ. Sci. Bp. Sect. Comput. 29, 141–156 (2008)
Székelyhidi, L.: Functional Equations on Hypergroups. World Scientific Publishing Co. Pte. Ltd., Hackensack, London (2012)
Székelyhidi, L.: On spectral synthesis in several variables. Adv. Oper. Theory 2(2), 179–191 (2017)
Székelyhidi, L.: Spherical spectral synthesis. Acta Math. Hung. 153(1), 120–142 (2017)
van Dijk, G.: Introduction to Harmonic Analysis and Generalized Gelfand Pairs. de Gruyter Studies in Mathematics. Walter de Gruyter & Co., Berlin (2009)
Wolf, J.: Spherical functions on Euclidean space. J. Funct. Anal. 239, 127–136 (2006)
Acknowledgements
The study was funded by Hungarian National Foundation for Scientific Research (OTKA) with Grant No. K111651.
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Fechner, Ż., Székelyhidi, L. Moment Functions on Affine Groups. Results Math 74, 5 (2019). https://doi.org/10.1007/s00025-018-0926-2
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DOI: https://doi.org/10.1007/s00025-018-0926-2