Problems in Function Theory and Operator Theory

申请人建议解决两组问题。第一个是申请人与Arcozzi（博洛尼亚），Sawyer（汉密尔顿）和Wick（亚特兰大）合作的延续。本组研究全纯函数空间的算子论泛函理论中的具体问题，全纯函数是位空间的子空间。经典的狄利克雷空间是一个基本的例子；德鲁里-阿文森-哈迪空间可能是最重要的例子。这些问题与经典的勒贝格空间子空间问题相似；关于插值，零集，乘数代数，冠等问题。然而，所需要的技术是完全不同的，涉及到容量理论，涉及到对全纯函数空间和包含势空间的离散模型的使用。第二组问题的工作是与Xiang Tang （St. Louis）合作，试图理解Rankin-Cohen括号的各种作用。括号是常系数双微分算子，其系数是组合数。这些操作符自然出现在各种环境中。它们普遍存在的一些结构原因现在已经被理解了，但是对于申请人特别感兴趣的某些上下文中括号的出现却知之甚少。拟议研究的一个目标是改善这种情况。在20世纪70年代和80年代，交换调和分析和函数理论在数学上取得了深刻的进展。这项研究是由一种愿望驱动的，即看看如何将几种非常有效但看似非常不同的观点结合起来，以更深入地了解波形的变换规则。这些观点的成功统一使长期存在的数学问题得以解决。改进的洞察力也导致了数据分析和信号处理方面的根本性创新；使用小波进行图像压缩是早期的成功，到目前为止，这些思想的后代已经成为许多计算机科学家和电气工程师工具包的标准组成部分。在此期间发展的理论工具和见解也使本建议中的问题触手可及。所考虑的具体问题具有直接的数学意义。为解决这些问题而开发的分析工具将再次拓宽和加深对波形如何分析和操纵的理解。关于Rankin-Cohen括号的问题是另一种类型。当非常复杂和非常优雅的表达式出现在几个看似不相关的环境中时，对许多数学家来说，一个令人信服的美学要求是找到潜在的原因。研究这类问题有一个伟大的传统，尽管答案通常很平凡，但有时却相当深刻。

The applicant proposes to work on two sets of problems. The first is a continuation of the applicant's collaboration with Arcozzi (Bologna), Sawyer (Hamilton), and Wick (Atlanta). This group is working on specific problems in the operator theoretic function theory of spaces of holomorphic functions which are subspaces of potential spaces. The classical Dirichlet space is a fundamental example; the Drury-Arveson-Hardy space is perhaps the most important example. The questions are similar to the questions classically considered for subspaces of Lebesgue spaces; questions about interpolation, zero sets, multiplier algebras, coronas, etc. However the techniques required are quite different, involving capacity theory and involving use of discrete models for both the space of holomorphic functions and the containing potential space. The work on a second set of problems is work in collaboration with Xiang Tang (St. Louis) trying to understand the various roles of the Rankin-Cohen brackets. The brackets are constant coefficient bidifferential operators whose coefficients are combinatorial numbers. These operators arise naturally in a wide range of settings. Some of the structural reasons for their ubiquity are now understood, but the occurrence of the brackets in certain contexts of particular interest to the applicant is poorly understood. A goal of the proposed research is to improve that situation.In the 1970s and 1980s there were profound mathematical advances at the interface of commutative harmonic analysis and function theory. The research was driven by a desire to see how several very productive, but seemingly very different, viewpoints could be used together to get a deeper understanding of transformation rules for waveforms. The successful unification of the viewpoints allowed resolution of long standing mathematical questions. The improved insight also led to fundamental innovations in data analysis and signal processing; using wavelets for image compression was an early success and by now the descendents of these ideas are a standard part of the toolkit of many computer scientist and electrical engineers. The theoretical tools and insights developed during that time also brought the questions in this proposal within reach. The specific questions being considered are of direct mathematical interest. The analytical tools being developed to work on the questions will, again, broaden and deepen the understanding of how waveforms can be analyzed and manipulated. The questions about the Rankin-Cohen brackets are of a different sort. When very complicated and very elegant expressions arise in several seemingly unrelated contexts there is, for many mathematicians, a compelling aesthetic imperative to find the underlying reason. There is a great tradition of working on such questions and although the answers are often mundane, sometimes they are quite profound.