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

Abstract

Product formulas of the type \[ u k ( θ ) u k ( ϕ ) = ∫ 0 π u k ( ξ ) D ( ξ , θ , ϕ ) d ξ {u_k}(\theta ){u_k}(\phi ) = \int _0^\pi {{u_k}(\xi )D(} \xi ,\theta ,\phi )\;d\xi \] are obtained for the eigenfunctions of a class of second order regular and regular singular Sturm-Liouville problems on [ 0 , π ] [0,\pi ] by using the Riemann integration method to solve a Cauchy problem for an associated hyperbolic differential equation. When D ( ξ , θ , ϕ ) 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 , π ] M[0,\pi ] becomes a Banach algebra with the functions u k ( ξ ) / u 0 ( ξ ) {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.