On symmetry of discrete polynomial hypergroups
HTML articles powered by AMS MathViewer
- by Marc-Olivier Gebuhrer and Ryszard Szwarc
- Proc. Amer. Math. Soc. 127 (1999), 1705-1709
- DOI: https://doi.org/10.1090/S0002-9939-99-04667-5
- Published electronically: February 11, 1999
- PDF | Request permission
Abstract:
Let $H$ be a discrete polynomial hypergoup on $\mathbb {N}$ with Plancherel measure $\mu .$ If the hypergroup $H$ is symmetric, the set of characters $\widehat {H}$ can be identified with a compact subset of the real line which contains the support of $\mu .$ We show that the lower and upper bounds of $\widehat {H}$ and $\operatorname {supp} \mu$ coincide. In particular, the trivial character belongs to the support of the Plancherel measure.References
- A. Hulanicki, Groups whose regular representation weakly contains all unitary representations, Studia Math. 24 (1964), 37–59. MR 191998, DOI 10.4064/sm-24-1-27-59
- A. Hulanicki, On positive functionals on a group algebra multiplicative on a subalgebra, Studia Math. 37 (1970/71), 163–171. MR 310547, DOI 10.4064/sm-37-2-163-171
- Robert I. Jewett, Spaces with an abstract convolution of measures, Advances in Math. 18 (1975), no. 1, 1–101. MR 394034, DOI 10.1016/0001-8708(75)90002-X
- Alan L. T. Paterson, Amenability, Mathematical Surveys and Monographs, vol. 29, American Mathematical Society, Providence, RI, 1988. MR 961261, DOI 10.1090/surv/029
- G. Pólya and G. Szegö, Aufgaben und Lehrsätze aus der Analysis. Band I: Reihen. Integralrechnung. Funktionentheorie, Die Grundlehren der mathematischen Wissenschaften, Band 19, Springer-Verlag, Berlin-New York, 1964 (German). Dritte berichtigte Auflage. MR 0170985
- Ryszard Szwarc, Convolution structures associated with orthogonal polynomials, J. Math. Anal. Appl. 170 (1992), no. 1, 158–170. MR 1184732, DOI 10.1016/0022-247X(92)90011-2
- Ryszard Szwarc, A lower bound for orthogonal polynomials with an application to polynomial hypergroups, J. Approx. Theory 81 (1995), no. 1, 145–150. MR 1323759, DOI 10.1006/jath.1995.1040
- Matthias Vogel, Spectral synthesis on algebras of orthogonal polynomial series, Math. Z. 194 (1987), no. 1, 99–116. MR 871222, DOI 10.1007/BF01168009
Bibliographic Information
- Marc-Olivier Gebuhrer
- Affiliation: Institut de Recherche Mathématique Avancée Université Louis Pasteur et C.N.R.S. 7, rue René Descartes 67084 Strasbourg Cedex France
- Ryszard Szwarc
- Affiliation: Institute of Mathematics Wrocław University pl. Grunwaldzki 2/4 50–384 Wroc- ław, Poland
- Received by editor(s): January 21, 1997
- Received by editor(s) in revised form: September 9, 1997
- Published electronically: February 11, 1999
- Additional Notes: This work has been partially supported by KBN (Poland) under grant 2 P03A 030 09. The work has been done while the second author was visiting IRMA, University Louis Pasteur in February 1994.
- Communicated by: J. Marshall Ash
- © Copyright 1999 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 127 (1999), 1705-1709
- MSC (1991): Primary 43A62, 42C05
- DOI: https://doi.org/10.1090/S0002-9939-99-04667-5
- MathSciNet review: 1476129