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

9404316施瓦茨该奖项支持对问题的数学研究，这些问题的解决方案将来自一元和多变量的正交多项式理论、超群和测量代数的技术组合。超群是相对较新的数学概念。当人们希望对定义在欧几里得空间的子集上的函数进行调和分析时，当集合不是群时，它们就出现了。如果集合上的度量形成一个超群或一些以基函数为特征的更一般的度量代数，这种结构的损失通常可以得到补偿。在前人研究以Sturm-Liouville方程的多项式或特征函数为特征的一维紧超群的基础上，本文将把结构定理推广到非紧超群上。还将研究n维空间上那些具有多项式特征标的超群和那些特征标满足微分方程的超群。超群的调和分析的发展，特别是通过它们的子群来理解超群的结构。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。在助学金的支持下，本科生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，最好地例证了综合的思想。一种是试图将复杂的问题分解成基本的组成部分。然后分析这些组件的基本特征。最后，通过组件的重新组合来重建解决方案。傅里叶级数和傅里叶变换是在这种情况下使用的工具的例子；一个是离散的，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)它们是在研究微分方程式时自然产生的；(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. ***