Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-24T08:49:51.007Z Has data issue: false hasContentIssue false
  • English
  • Français

Positive Definite and Related Functions on Hypergroups

Published online by Cambridge University Press:  20 November 2018

Walter R. Bloom  and
Paul Ressel
Walter R. Bloom
Affiliation:
School of Mathematical and Physical Sciences Murdoch UniversityPerth WA 6150, Australia
Paul Ressel
Affiliation:
Mathematisch-Geog raphische Fakultät Katholische Universität EichstättD-8078 Eichstätt Federal Republic of Germany
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we make use of semigroup methods on the space of compactly supported probability measures to obtain a complete Lévy-Khinchin representation for negative definite functions on a commutative hypergroup. In addition we obtain representation theorems for completely monotone and completely alternating functions. The techniques employed here also lead to considerable simplification of the proofs of known results on positive definite and negative definite functions on hypergroups.

Keywords

60B0543A1043A35
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 43 , Issue 2 , 01 April 1991 , pp. 242 - 254
Copyright
Copyright © Canadian Mathematical Society 1991

References

1. Berg, Christian, Reus Christensen, Jens Peter and Ressel, Paul, Harmonic analysis on semigroups. Graduate Texts in Mathematics, 100, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1984.Google Scholar
2. Bloom, Walter R. and Herbert Heyer, Characterisation of potential kernels of transient convolution semigroups on a commutative hypergroup. Probability measures on groups IX (Proc. Conf., Oberwolfach Math. Res. Inst., Oberwolfach 1988), Lecture Notes in Math., 1379, Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, 1989, 21-35.Google Scholar
3. Buchwalter, Henri, Les fonctions de Levy existent!, Math. Ann. 274(1986), 31–34.Google Scholar
4. Hewitt, Edwin and Ross, Kenneth A., Abstract harmonic analysis, vol II. Die Grundlehren der mathematischen Wissenschaften, 152, Springer-Verlag, Berlin, Heidelberg, New York, 1970.Google Scholar
5. Heyer, Herbert, Probability measures on locally compact groups. Ergebnisse der Mathematik und ihrer Grenzgebiete, 94, Springer-Verlag, Berlin, Heidelberg, New York, 1977.Google Scholar
6. Jewett, Robert I., Spaces with an abstract convolution of measures, Adv. in Math. 18(1975), 1–101.Google Scholar
7. Lasser, Rupert, Orthogonal polynomials and hypergroups, Rend. Mat. (Series VII)3(1983), 185–209.Google Scholar
8. Lasser, Rupert, Convolution semigroups on hypergroups, Pacific J. Math. 127(1987), 353–371.Google Scholar
9. Ressel, Paul, Integral representations on convex semigroups, Math. Scand. 61(1987), 93–111.Google Scholar
10. Spector, René, Mesures invariantes sur les hypergroupes, Trans. Amer. Math. Soc. 239(1978), 147–165.Google Scholar
11. Voit, Michael, Positive characters on commutative hypergroups and some applications, Math. Z. 198( 1988), 405–421.Google Scholar
12. Voit, Michael, Negative definite functions on commutative hypergroups. Probability measures on groups IX(Proc. Conf., Oberwolfach Math. Res. Inst., Oberwolfach 1988), Lecture Notes in Math., 1379, Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, 1989, 376–388.Google Scholar