Problems in Function Theory and Operator Theory

这个项目有两个组成部分。第一部分是对对称Fock空间的算子理论和函数理论的研究。这位首席研究员和他的合作者最近开发了该空间的Carleson测度的几何特征。他们相信，这一结果，连同在获得它时开发的离散化技术，将为主要的直接目标提供有效的工具，即描述该Fock空间的内插序列。第二部分研究了Hardy空间和Bergman空间张量积的直和分解与Rankin-Cohen括号运算之间的关系。括号和分解都使用相同的双线性微分算子，即横向向量来实现。因此，很自然地推测，由括号在自同构形式的分次空间上诱导的代数结构(结合的、非对易的乘积结构)反映在张量积分解中的一个新的代数结构上。这个项目的第一部分，研究对称Fock空间，是三十年来在函数论和调和分析中发展起来的数学程序的一部分--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，该程序还在信号处理技术和其他数据分析领域带来了重大进步。小波分析是这个程序最著名的产品，但还有许多其他的产品。关于Fock空间的工作完全符合这一传统。它将探索一种在理论数学和数值应用中非常有效的特殊离散化技术如何适应更复杂的几何环境。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，是对观察到的相同的复杂计算结构(“跨向量”)在这两个迄今不相关的背景下出现的回应。数学中的一个基本原理是，这种“意外”几乎总是第一个迹象，表明在被视为无关的领域之间存在意想不到的、有时甚至是深层次的联系。申请者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.