Mathematical Sciences: Research in Orthogonal Polynomials and Hypergroups

摘要 Schwartz，Connett Schwartz和Connett将继续研究谐波分析 欧氏空间子集H上函数族的展开式 n-空间，当H不是群时。这种群体结构的缺乏， 如果H上的测度形成一个超群或某些超群， 以基函数为特征标的更一般测度代数。 福尔斯的研究主要包括以下几个方面：（1）扩展 提出者关于多项式超群分类的工作 （2）继续研究了超群与超群的关系， 线性（偏）微分算子，（3）发展抽象 （4）研究了信号中出现的球面波函数的调和分析 加工和其他应用，（5）研究性能， 具有正交多项式特征标的非超群测度代数。 （6）学生研究项目。 众所周知，任何声音，无论多么复杂， 可以被分解成一系列纯音，例如 用音叉这种对复杂物体的分解 转换成简单的积木（如音叉音调）是在 许多现代技术的核心，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 在气候和其他自然现象的数学建模中起着重要作用 现象以及数学中的许多重要问题。 在许多这样的情况下， 比纯粹的音叉音调复杂。这样做的目的 该项目是研究和了解各种 这些系统的积木，并发现新的系统，可能被证明是有用的，在未来。

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.