Mathematical Sciences: Research in Orthogonal Polynomials and Hypergroups

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ABSTRACT Schwartz, Connett Schwartz and Connett will continue to study the harmonic analysis of expansions in a family of functions on a subset H of Euclidean n-space when H is not a group. This lack of group structure can often be compensated if the measures on H form a hypergroup or some more general measure algebra with the basis functions as characters. The research proposed here falls into several areas: (1) Extend the proposers' work on the classification of hypergroups with polynomial characters to those with multivariable polynomials as characters, (2) continue to investigate the relation between hypergroups and linear (partial) differential operators, (3) develop the abstract harmonic analysis of hypergroups, especially in the light of the behavior of the maximum subgroup, (4) investigate the harmonic analysis of the spheroidal wave functions which arise in signal processing and other applications, (5) study the properties of non-hypergroup measure algebras with orthogonal polynomial characters. and (6) a student research project. It is well known that any sound, no matter how complicated, can be decomposed into a series of pure tones, such as those produced by a tuning fork. This kind of decomposition of a complicated object into simple building blocks (such as the tuning fork tones) is at the heart of many modern technologies such as signal compression and high definition TV, cat scans, nmr scans and image processing. It also plays a role in mathematical modeling of climate and other physical phenomena as well as for many important problems inside of mathematics. In many of these situations, the basic building blocks are more complicated than the pure tuning fork tones. The purpose of this project is to study and understand the properties of a variety of these systems of building blocks and to discover new systems that may prove to be useful in the future.