Research in Harmonic Analysis

本计画包含调和分析研究的几个主要部分：第一，主要研究者将研究对称空间上函数的球面变换，特别是球面乘子与球面乘子变换的定义与研究。 其次，主要研究者将研究有界对称域的热核。 他将找到所有有界对称域的热核的统一的显式公式。接下来，首席研究员将继续他的研究在有界对称域上的全纯函数空间的N-宽度。 最后，主要研究者将继续研究贝塞尔函数的解析延拓和埃尔米特对称空间的自同构群的全纯离散级数表示。总之，所有这四个项目都是对称空间分析的一部分，这是一个不断发展的数学研究领域。发展于十七世纪的微积分和经典数学分析是现代数学、物理学、科学、工程和先进技术的基础。由于其重要性，对称空间的分析在多元统计学、量子力学甚至分子化学等学科中都有应用。近十年来，主要研究者一直致力于这一领域的研究，并取得了一些重要成果。在总结前人研究成果的基础上，结合本领域的新进展，期望能有更多的研究成果。

AbstractDingThe project consists of several major components of research in harmonic analysis.First, the Principal Investigator will study the spherical transform of functions on symmetric spaces, in particular, he will define and study the spherical multipliers and spherical multiplier transformations. Secondly, the Principal Investigator will study heat kernels of bounded symmetric domains. He will find a uniform, explicit formula for heat kernels of all bounded symmetric domains. Next, the Principal Investigator will continue his investigation on the N-widths of spaces of holomorphic functions on bounded symmetric domains. Finally, the Principal Investigator will continue to study analytic continuation of Bessel functions and holomorphic discrete series representations of automorphism groups of Hermitian symmetric spaces. In summary, all of these four projects are parts of ANALYSIS ON SYMMETRIC SPACES, a growing area of mathematical research. Calculus and classical mathematical analysis, developed in the seventeenth century, have been foundations of modern mathematics, physics, science,engineering and advanced technology. Because of its importance, analysis on symmetric spaces has applications in such disciplines as multivariate statistics, quantum mechanics, and even molecular chemistry. In last ten years, the Principal Investigator has been woking on this area, and has obtained some important results. Base on results of previous research and new developments in this area, more results are expected to come.