Problems in Function Theory and Operator Theory

该项目有两个组成部分。第一部分研究对称Fock空间的算子理论和函数理论。首席研究员和他的合作者最近开发了该空间的Carleson措施的几何特征。 他们相信，这一结果，再加上离散化技术的发展，在获得它，将提供有效的工具，为主要的直接目标，这是描述该Fock空间的插值序列。第二部分是研究Rankin-Cohen括号运算与哈代和Bergman空间张量积的直和分解之间的关系。括号和分解都使用相同的双线性微分算子transvectant来实现。因此，很自然地推测，在自守形式的分次空间上由括号（一种结合的、非交换的乘积结构）诱导的代数结构，在张量积分解中被一种新的代数结构所反映。该项目的第一个组成部分，研究对称Fock空间，是一个数学计划的一部分，该计划已经在函数论和调和分析中发展了三十年-使用离散化技术来研究连续现象。除了对理论数学的贡献外，该计划还导致了信号处理技术和其他数据分析领域的重大进步。小波分析是这个程序最著名的产品，但还有很多其他的。 Fock空间的工作完全符合这一传统。这将是一个探索如何一个特定的离散化技术，这是已知的是非常有效的理论数学和数值应用可以适应一个更复杂的几何设置。该项目的第二个组成部分，研究Rankin-Cohen括号和某些张量积的分解之间的关系，是对观察到相同的复杂计算结构（"transvectants"）出现在这两个迄今为止无关的上下文中的反应。数学中的一个基本原则是，这种"偶然事件"几乎总是最初的迹象，表明在被视为不相关的领域之间存在着意想不到的、有时是深刻的联系。申请人建议确定这就是这里的情况，并发展这些联系所提出的新见解。

This project has two components. The first is a study of the operator theory and function theory of the symmetric Fock space. The principal investigator and his collaborators have recently developed a geometric characterization of the Carleson measures for that space. They believe that this result, together with the discretization techniques developed in obtaining it, will provide effective tools for the major immediate goal, which is to describe the interpolating sequences for this Fock space. The second component is the study of the relation between the Rankin-Cohen bracket operation and the direct sum decomposition of tensor products of Hardy and Bergman spaces. Both the bracket and the decomposition are implemented using the same bilinear differential operator, the transvectant. Hence it is natural to speculate that the algebraic structure induced on the graded space of automorphic forms by the bracket (an associative, noncommutative product structure) is mirrored by a new algebraic structure in the tensor product decomposition. The principal investigator proposes to identify that structure and develop its algebraic and analytic properties.The first component of this project, studying the symmetric Fock space, is part of a mathematical program that has evolved in function theory and harmonic analysis for thirty years -- the use of discretization techniques to study continuous phenomena. Beyond its contribution to theoretical mathematics, this program has led to major advances in signal processing techniques and other areas of data analysis. Wavelet analysis is the best known product of this program, but there are many others. The work on the Fock space is squarely in that tradition. It will be an exploration of how a particular discretization technique that is known to be very effective in theoretical mathematics and in numerical applications can be adapted to a much more sophisticated geometric setting. The second component of the project, the study of the relation between the Rankin-Cohen bracket and the decomposition of certain tensor products, is a response to the observation that the same complicated computational constructs ("transvectants") show up in these two, so far, unrelated contexts. A basic principle in mathematics is that such "accidents" are almost always the first indication of unexpected, and sometimes deep, connections between what had been seen as unrelated areas. The applicant proposes to establish that that is the situation here and to develop the new insights suggested by those connections.