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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Convolution and hypergroup structures associated with a class of Sturm-Liouville systems


Authors: William C. Connett, Clemens Markett and Alan L. Schwartz
Journal: Trans. Amer. Math. Soc. 332 (1992), 365-390
MSC: Primary 43A10; Secondary 34B24, 47B38
MathSciNet review: 1053112
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Abstract: Product formulas of the type

$\displaystyle {u_k}(\theta ){u_k}(\phi ) = \int_0^\pi {{u_k}(\xi )D(} \xi ,\theta ,\phi )\;d\xi $

are obtained for the eigenfunctions of a class of second order regular and regular singular Sturm-Liouville problems on $ [0,\pi ]$ by using the Riemann integration method to solve a Cauchy problem for an associated hyperbolic differential equation.

When $ D(\xi ,\theta ,\phi )$ is nonnegative (which can be guaranteed by a simple restriction on the differential operator of the Sturm-Liouville problem), it is possible to define a convolution with respect to which $ M[0,\pi ]$ becomes a Banach algebra with the functions $ {u_k}(\xi )/{u_0}(\xi )$ as its characters. In fact this measure algebra is a Jacobi type hypergroup. It is possible to completely describe the maximal ideal space and idempotents of this measure algebra.


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DOI: https://doi.org/10.1090/S0002-9947-1992-1053112-6
Keywords: Measure algebras, convolutions, hypergroups, product formulas, Sturm-Liouville problems, characters, eigenfunctions
Article copyright: © Copyright 1992 American Mathematical Society