Problems in Function Theory and Operator Theory

这个项目有两个组成部分。首先研究了对称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测度的几何特征。他们认为，这一结果，以及在获得它的过程中开发的离散化技术，将为描述Fock空间的插值序列这一主要直接目标提供有效的工具。第二部分是研究Rankin-Cohen括号运算与Hardy和Bergman空间张量积的直接和分解之间的关系。括号和分解都是使用相同的双线性微分算子，即透变算子来实现的。因此，我们可以很自然地推测，在自同构形式的梯度空间上由括号（一种结合的、非交换的积结构）诱导出的代数结构在张量积分解中被一种新的代数结构所反映。首席研究员建议确定该结构并发展其代数和解析性质。该项目的第一个组成部分，研究对称的Fock空间，是在函数理论和谐波分析中发展了30年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之外，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中最著名的产品，但还有许多其他的。福克空间的作品完全符合这一传统。它将探索如何将一种在理论数学和数值应用中非常有效的特定离散化技术应用于更复杂的几何环境。该项目的第二个组成部分是研究Rankin-Cohen括号与某些张量积分解之间的关系，这是对观察到的相同的复杂计算结构（“透量”）在这两个迄今为止不相关的上下文中出现的回应。数学的一个基本原则是，这种“意外”几乎总是在看似不相关的领域之间出现意想不到的、有时甚至是深刻的联系的第一个迹象。申请人建议确定这就是这里的情况，并发展这些联系所暗示的新见解。

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.