Harmonic Analysis and Special Functions

摘要Richards Richards将继续他关于定义在矩阵空间和对称锥上的超几何函数的研究计划。我们将特别关注对称锥上的算子值超几何函数理论。在项目的第二部分，主要研究人员将继续研究这些特殊函数以及调和分析和表示理论中的一般技术在多元统计、全正性、概率不等式和相关组合问题中的应用。理查兹对研究问题现代分析感兴趣，这是数学的一个子领域，原因有两个。首先，现代分析是数学的一个重要分支，与天文学、化学、物理学、统计学以及数学的其他领域有着基本的联系。理查兹在现代分析方面的主要专长是研究新的超几何函数类，这些超几何函数类推广了欧拉、高斯、贝塞尔、拉马努扬和许多其他著名科学家发展的经典理论。第二，主要研究人员对这些新类型的超几何函数感兴趣，因为它们在多元统计中具有基本的重要性。该研究项目将研究这些新的超几何函数及其性质，目的是为尚未解决的统计问题找到解决方案，这些问题对医学和环境领域的应用研究人员很重要。

ABSTRACT Richards Richards will continue his program of research on hypergeometric functions defined on matrix spaces and on symmetric cones. Special attention will be paid to the theory of operator-valued hypergeometric functions on the symmetric cones. In the second part of the project the principal investigator will continue his research on applications of these special functions, and general techniques in harmonic analysis and representation theory, to the areas of multivariate statistics, total positivity, probability inequalities, and related combinatorial problems. Richards is interested in research problems Modern Analysis, a subfield of mathematics, for two reasons. First, Modern Analysis is an important branch of mathematics with fundamental links to astronomy, chemistry, physics, statistics, as well as to other areas of mathematics. Richards primary expertise in Modern Analysis is in the study of new classes of hypergeometric functions which generalize the classical theory developed by Euler, Gauss, Bessel, Ramanujan and many other famous scientists. Second, the principal investigator is interested in these new types of hypergeometric functions because of their fundamental importance in multivariate statistics. The research program will study these new hypergeometric functions and their properties with the aim of finding solutions to unsolved statistical problems which are of importance to applied researchers in medical and environmental fields.