Mathematical Sciences: Orthogonal Polynomals in One and Several Variables, Hypergroups, and Related Measure Algebras

小行星9404316 该奖项支持数学研究的问题，其解决方案将来自一个和多个变量，超群和测度代数的正交多项式理论的技术组合。 超群是相对较新的数学概念。 它们出现时，人们希望执行的调和分析功能定义的子集的欧几里德空间时，集不组。 如果集合上的测度形成一个超群或以基函数为特征的更一般的测度代数，这种结构的损失通常可以得到补偿。 本文在前人研究特征为Sturm-Liouville方程的多项式或特征函数的紧的一维超群的基础上，将结构定理推广到非紧超群. 研究了n维空间中具有多项式特征的超群和具有满足微分方程特征的超群。 调和分析的发展，特别是通过子群理解超群的结构也将进行。 最后，研究了特征标集由球面波函数和其他特殊函数组成的非超群测度代数的性质和调和分析。 本科生在补助金的支持下，将参与开发相关符号计算的示例和计算机检查。 调和分析结合了这些数学元素，最好地体现了综合的思想。 一种是试图将复杂的问题分解为基本的组成部分。 然后分析这些组件的基本特性。 最后，通过组件的重组来重构解决方案。 傅立叶级数和傅立叶变换是在这种情况下使用的工具的例子;一个离散，另一个表示连续的分解。 正交多项式在调和分析的某些部分中起着重要的作用，因为（a）它们在微分方程的研究中自然出现（B）它们可以用来表示大类函数（c）它们通常很容易计算。 ***

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. ***