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

Connett, William C.; Markett, Clemens; Schwartz, Alan L.

doi:10.1090/S0002-9947-1992-1053112-6

Convolution and hypergroup structures associated with a class of Sturm-Liouville systems
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by William C. Connett, Clemens Markett and Alan L. Schwartz PDF

Trans. Amer. Math. Soc. 332 (1992), 365-390 Request permission

Abstract:

Product formulas of the type \[ {u_k}(\theta ){u_k}(\phi ) = \int _0^\pi {{u_k}(\xi )D(} \xi ,\theta ,\phi )\;d\xi \] 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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