Mathematical Sciences: Research in Orthogonal Polynomials and Hypergroups

当H不是群时，Schwartz、Connett Schwartz和Connett将继续研究欧几里得n-空间的子集H上的函数族的展开的调和分析。如果H上的测度形成一个超群或一些更一般的以基函数为特征的测度代数，则这种群结构的缺失通常可以得到弥补。本文的研究内容包括以下几个方面：(1)将前人关于特征标为多项式的超群的分类工作推广到特征标为多元多项式的超群的分类；(2)继续研究超群与线性(偏)微分算子之间的关系；(3)发展超群的抽象调和分析，特别是结合最大子群的性质；(4)研究信号处理和其它应用中出现的球面波函数的调和分析；(5)研究具有正交多项式特征标的非超群测度代数的性质。和(6)学生研究项目。众所周知，任何声音，无论多么复杂，都可以分解成一系列纯音，比如音叉发出的纯音。这种将复杂的物体分解成简单的积木(如音叉音调)的方式是许多现代技术的核心，例如信号压缩和高清晰度电视、CAT扫描、核磁共振扫描和图像处理。它还在气候和其他物理现象的数学建模以及数学中的许多重要问题中发挥作用。在许多这样的情况下，基本构件比纯音叉音要复杂得多。这个项目的目的是研究和了解各种构建块系统的特性，并发现可能被证明在未来有用的新系统。

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.