Mathematical Sciences: Convolution Algebras and Sturm Liouville Problems

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This project seeks to blend mathematical research on the harmonic analysis of eigenfunctions of a Sturm-Liouville differential equation with the algebraic construct of a hypergroup. The hypergroup consists of an algebra of measures supported on the interval on which the equation is defined. There is a converse direction as well: a one-dimensional hypergroup defines a Sturm-Liouville problem and the characters in the measure algebra are normalized eigenfunctions of the problem. The work to be carried out falls into four areas: first, to continue exploiting the relationship between hypergroup measure algebras and the measure algebras that have Sturm-Liouville eigenfunctions as their characters. Second, investigate structural questions about hypergroups themselves. Third, the harmonic analysis of eigenfunction expansions will carried out and fourth, a student research project to develop the computational and graphic software in support of the first three areas is planned. Results of this work are expected to have applications to probability theory, orthogonal polynomials and the theory of special functions.