Research in Harmonic Analysis

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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.