Analysis of a class of probability preserving measure algebras on compact intervals

Abstract

The measure algebras of the title are those which are also hypergroups with some regularity conditions. Examples include the convolutions associated with Jacobi polynomial series and Fourier Bessel series. It is shown here that there is a one-to-one correspondence between these hypergroups and a class of Sturm-Liouville problems which have the characters of the hypergroup as eigenfunctions. The interplay between these two characterizations allows a detailed analysis which includes a Hilb-type formula for the characters and asymptotic estimates for the Plancherel measure and the eigenvalues of the associated Sturm-Liouville problem.