9404316 Schwartz This award supports mathematical research on problems whose solutions will derive from a combination of techniques from the theory of orthogonal polynomials in one and several variables, hypergroups and measure algebras. Hypergroups are relatively new mathematical constructs. They arise when one wishes to perform a harmonic analysis of functions defined on subsets of Euclidean space when the sets are not groups. This loss of structure can often be compensated if the measures on the set form a hypergroup or some more general measure algebra with basis functions as characters. Building on the previous work which focused on compact one-dimensional hypergroups whose characters are polynomials or eigenfunctions of Sturm-Liouville equations, efforts now will be made to extend the structure theorems to non- compact hypergroups. Work will also be done investigating those hypergroups on n-dimensional space that have polynomials characters and those which have characters which satisfy differential equations. The development of harmonic analysis of hypergroups, and, in particular, understanding the structure of hypergroups through their subgroups will also be carried out. Finally, the properties and harmonic analysis of non-hypergroup measure algebras with character sets consisting of spherical wave functions and other special functions will be studied. Undergraduates, supported by the grant, will participate in developing examples and computer checks of relevant symbolic computations. Harmonic analysis combines those elements of mathematics best exemplifying the ideas of synthesis. One seeks to decompose complex problems into fundamental components. These components are then analyzed for their basic characteristics. Finally, the solution is reconstructed through a recombination of the components. The Fourier series and Fourier transform are examples of tools used in this context; one discrete , the other representing a continuous d ecomposition. Orthogonal polynomials play an important role in certain segments of harmonic analysis because (a) they arise naturally in the study of differential equations (b) they can be used to represent large classes of functions and (c) they are often easily computable. ***
