The focus of this project is a generalized harmonic analysis involving orthogonal functions on sets in higher dimensional Euclidean space. The novel feature of the research is that the sets under consideration do not form a group. This lack of algebraic structure can often be compensated for by the existence of an adequate algebraic structure in the measure algebra supported on the set. The structures under consideration here are referred to as hypergroups in which the algebraic operation is convolution. If the elements of a basis of orthogonal functions act as homomorphisms of the hypergroup, then one gets a viable mathematical structure. For example, the initial one- dimensional work in the area used eigenfunctions of a Sturm- Liouville problem as a basis. The current project seeks to extend existing structure theorems for compact one-dimensional hypergroups to the non-compact case. Other work will focus on characterizing the compactly supported hypergroups in higher dimensions, developing a harmonic analysis of the disk polynomials and the spheroidal wave functions and analyzing the structure of specific families of non-hypergroup measure algebras. The hypergroup concept, abstract as it is, has its roots in probability theory, orthogonal polynomials and the theory of special functions. Particular applications include the explicit representations for annular spheroidal wave functions even though the functions cannot be given in closed form and the development of hypergroup interpretations of characteristic functions in probability theory.
