Skip to main content
SpringerLink
Account
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Results in Mathematics
  3. Article

Moment Functions on Affine Groups

  • Open access
  • Published: 17 November 2018
  • volume 74, Article number: 5 (2019)
Download PDF

You have full access to this open access article

Results in Mathematics Aims and scope Submit manuscript
Moment Functions on Affine Groups
Download PDF
  • Żywilla Fechner  ORCID: orcid.org/0000-0001-7412-65441 &
  • László Székelyhidi2 
  • 340 Accesses

  • 1 Citation

  • Explore all metrics

Cite this article

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.

Article PDF

Download to read the full article text

Use our pre-submission checklist

Avoid common mistakes on your manuscript.

References

  1. Aczél, J.: Functions of binomial type mapping groupoids into rings. Math. Z. 154, 115–124 (1977)

    Article  MathSciNet  Google Scholar 

  2. Artzy, R.: Linear Geometry. Addison-Wesley, Reading (1965)

    MATH  Google Scholar 

  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)

    Book  Google Scholar 

  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)

    MATH  Google Scholar 

  5. Fechner, Ż., Székelyhidi, L.: Sine functions on hypergroups. Arch. Math. (Basel) 106(4), 371–382 (2016)

    Article  MathSciNet  Google Scholar 

  6. Fechner, Ż., Székelyhidi, L.: Functional equations on double coset hypergroups. Ann. Funct. Anal. 8(3), 411–423 (2017)

    Article  MathSciNet  Google Scholar 

  7. Fechner, Ż., Székelyhidi, L.: Sine and cosine equations on hypergroups. Banach J. Math. Anal. 11(4), 808–824 (2017)

    Article  MathSciNet  Google Scholar 

  8. Hermann, P.: Representations of double coset hypergroups and induced representations. Manuscr. Math. 88(1), 1–24 (1955)

    Article  MathSciNet  Google Scholar 

  9. Hewitt, E., Ross, K.A.: Abstract Harmonic Analysis. Vol. I. Fundamental Principles of Mathematical Sciences, vol. 115. Springer, Berlin (1979)

    Book  Google Scholar 

  10. Lyndon, R.C.: Groups and Geometry. London Mathematical Society Lecture Note Series, vol. 101. Cambridge University Press, Cambridge (1985)

    Book  Google Scholar 

  11. Orosz, Á., Székelyhidi, L.: Moment functions on polynomial hypergroups in several variables. Publ. Math. Debr. 65(3–4), 429–438 (2004)

    MathSciNet  MATH  Google Scholar 

  12. Orosz, Á., Székelyhidi, L.: Moment functions on polynomial hypergroups. Arch. Math. 85(2), 141–150 (2005)

    Article  MathSciNet  Google Scholar 

  13. Orosz, Á., Székelyhidi, L.: Moment functions on Sturm–Liouville hypergroups. Ann. Univ. Sci. Bp. Sect. Comput. 29, 141–156 (2008)

    MathSciNet  MATH  Google Scholar 

  14. Székelyhidi, L.: Functional Equations on Hypergroups. World Scientific Publishing Co. Pte. Ltd., Hackensack, London (2012)

    MATH  Google Scholar 

  15. Székelyhidi, L.: On spectral synthesis in several variables. Adv. Oper. Theory 2(2), 179–191 (2017)

    MathSciNet  MATH  Google Scholar 

  16. Székelyhidi, L.: Spherical spectral synthesis. Acta Math. Hung. 153(1), 120–142 (2017)

    Article  MathSciNet  Google Scholar 

  17. van Dijk, G.: Introduction to Harmonic Analysis and Generalized Gelfand Pairs. de Gruyter Studies in Mathematics. Walter de Gruyter & Co., Berlin (2009)

    Book  Google Scholar 

  18. Wolf, J.: Spherical functions on Euclidean space. J. Funct. Anal. 239, 127–136 (2006)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

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

Author information

Authors and Affiliations

  1. Institute of Mathematics, Lodz University of Technology, ul. Wólczańska 215, 90-924, Łódź, Poland

    Żywilla Fechner

  2. University of Debrecen, Debrecen, Hungary

    László Székelyhidi

Authors
  1. Żywilla Fechner
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. László Székelyhidi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Żywilla Fechner.

Rights and permissions

Open Access This 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.

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fechner, Ż., Székelyhidi, L. Moment Functions on Affine Groups. Results Math 74, 5 (2019). https://doi.org/10.1007/s00025-018-0926-2

Download citation

  • Received: 17 January 2018

  • Accepted: 08 November 2018

  • Published: 17 November 2018

  • DOI: https://doi.org/10.1007/s00025-018-0926-2

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Hypergroup
  • generalized moment function
  • affine group
  • spherical functions

Mathematics Subject Classification

  • 20N20
  • 43A62
  • 39B99
Use our pre-submission checklist

Avoid common mistakes on your manuscript.

Advertisement

search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

Not affiliated

Springer Nature

© 2023 Springer Nature