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Generalized hypergroups and orthogonal polynomials

Part of: Abstract harmonic analysis Commutative Banach algebras and commutative topological algebras

Published online by Cambridge University Press:  09 April 2009

Rupert Lasser ,
Josef Obermaier  and
Holger Rauhut
Rupert Lasser
Affiliation:
GSF—National Research Center for Environment and HealthInstitute of Biomathematics and BiometryIngolstädter Landstrasse 1D-85764 [email protected]@gsf.de
Josef Obermaier
Affiliation:
University of ViennaFaculty of MathematicsNuHAGNordbergstr. 15A-1090 [email protected]
Holger Rauhut
Affiliation:
University of ViennaFaculty of MathematicsNuHAGNordbergstr. 15A-1090 [email protected]
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Abstract

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The concept of semi-bounded generalized hypergroups (SBG hypergroups) is developed. These hypergroups are more special than generalized hypergroups introduced by Obata and Wildberger and more general than discrete hypergroups or even discrete signed hypergroups. The convolution of measures and functions is studied. In the case of commutativity we define the dual objects and prove some basic theorems of Fourier analysis. Furthermore, we investigate the relationship between orthogonal polynomials and generalized hypergroups. We discuss the Jacobi polynomials as an example.

Keywords

generalized hypergroupsemi-bounded generalized hypergroupbounded generalized hypergroupssigned hypergroupdiscrete hypergroupconvolutiondual objectFourier transformorthogonal polynomialsJacobi polynomials.

MSC classification

Secondary: 43A62: Hypergroups 43A99: None of the above, but in this section 46J10: Banach algebras of continuous functions, function algebras 33C80: Connections with groups and algebras, and related topics
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 82 , Issue 3 , June 2007 , pp. 369 - 394
Copyright
Copyright © Australian Mathematical Society 2007

References

[1]Askey, R. and Gasper, G., ‘Linearization of the product of Jacobi polynomials III’, Canad. J. Math. 23 (1971), 332–338.CrossRefGoogle Scholar
[2]Bloom, W. R. and Heyer, H., Harmonic analysis of probability measures on hypergroup (de Gruyter, Berlin, 1995).CrossRefGoogle Scholar
[3]Chihara, T. S., An introduction to orthogonal polynomials (Gordon and Breach, New York, 1978).Google Scholar
[4]Dunkl, C., ‘The measure algebra of a locally compact hypergroup’, Trans. Amer. Math. Soc. 179 (1973), 331–348.CrossRefGoogle Scholar
[5]Filbir, F., Lasser, R. and Obermaier, J., ‘Summation kernels for orthogonal polynomials’, in: Handbook on analytic-computational methods in applied mathematics (ed. Anastassiou, G.) (Chapman and Hall, Boca Raton, 2000) pp. 709–749.Google Scholar
[6]Gasper, G., ‘Linearization of the product of Jacobi polynomials II’, Canad. J. Math. 22 (1970), 582–593.CrossRefGoogle Scholar
[7]Jewett, R. I., ‘Spaces with an abstract convolution of measures’. Adv. Math. 18 (1975), 1–101.CrossRefGoogle Scholar
[8]Lasser, R., ‘Orthogonal polynomials and hypergroups’. Rend. Mat. (7) 3 (1983), 185–209.Google Scholar
[9]Lasser, R. and Leitner, M., ‘Stochastic processes indexed by hypergroups I’, J. Theoret. Probab. 2 (1989), 301–311.CrossRefGoogle Scholar
[10]Lasser, R. and Obermaier, J., ‘On the convergence of weighted Fourier expansions’, Acta. Sci. Math. (Szeged) 61 (1995), 345–355.Google Scholar
[11]Leitner, M., ‘Stochastic processes indexed by hypergroups II’. J. Theoret. Probab. 4 (1991), 321–331.CrossRefGoogle Scholar
[12]Obata, N. and Wildberger, N. J., ‘Generalized hypergroups and orthogonal polynomials’, Nagoya Math. J. 142 (1996), 67–93.CrossRefGoogle Scholar
[13]Ross, K. A.. ‘Signed hypergroups — a survey’, Contemp. Math. 183 (1995), 319–329.CrossRefGoogle Scholar
[14]Spector, R., Apercu de la Théorie des Hypergroupes, Lecture Notes in Math.. Vol. 497 (Analyse Harmonique sur les Groupes de Lie, Sem. Nancy-Strasbourg 1973–1975) (Springer, Berlin, 1975).Google Scholar
[15]Szegö, G., Orthogonal polynomials (Amer. Math. Soc., New York, 1959).Google Scholar
[16]Werner, D., Funktionalanalysis (Springer, Berlin, 1995).Google Scholar