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

9404316 Schwartz该奖项支持对一些问题的数学研究，这些问题的解决方案将来自于单变量和多变量正交多项式理论、超群和测度代数的技术组合。超群是相对较新的数学结构。当人们希望对定义在欧几里德空间的子集上的函数进行调和分析时，当这些集合不是群时，它们就出现了。如果集合上的测度形成了一个超群或以基函数为特征的更一般的测度代数，这种结构上的损失通常可以得到补偿。在先前研究以Sturm-Liouville方程的多项式或本征函数为特征的紧一维超群的基础上，现在将努力将结构定理推广到非紧超群。我们还将研究n维空间上那些具有多项式特征和那些具有满足微分方程特征的超群。发展超群的调和分析，特别是通过它们的子群来理解超群的结构。最后，研究了由球波函数和其他特殊函数组成的特征集的非超群测度代数的性质及其调和分析。本科生将参与开发相关符号计算的示例和计算机检查。谐波分析结合了最能体现综合思想的数学元素。人们试图把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性。最后，通过组件的重组重构解。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，另一个是连续的。正交多项式在调和分析的某些部分起着重要的作用，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的函数类(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. ***