Problems in Function Theory and Operator Theory

申请人提议处理两组问题。第一个是申请人与Arcozzi（博洛尼亚），索耶（汉密尔顿）和威克（亚特兰大）合作的延续。这个小组的工作具体问题的算子理论的功能理论空间的全纯函数是子空间的潜在空间。经典的狄利克雷空间是一个基本的例子;德鲁里-阿维森-哈代空间也许是最重要的例子。这些问题是类似的问题经典考虑的子空间的勒贝格空间;问题的插值，零集，乘子代数，冠等，但所需的技术是完全不同的，涉及能力理论和涉及使用离散模型的空间全纯函数和包含潜在的空间。第二组问题的工作是与Xiang Tang（圣路易斯）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。这些操作符在广泛的环境中自然出现。它们普遍存在的一些结构原因现在已经被理解，但是在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。在20世纪70年代和80年代，在交换调和分析和函数论的界面上有了深刻的数学进步。这项研究是由一个愿望，看看几个非常富有成效的，但似乎非常不同的观点可以一起使用，以获得更深入的了解波形的转换规则。 成功的统一的观点允许解决长期存在的数学问题。洞察力的提高还带来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取得了早期成功，到目前为止，这些想法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部分。在此期间开发的理论工具和见解也使该提案中的问题触手可及。正在考虑的具体问题是直接的数学兴趣。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的。当非常复杂和非常优雅的表达式出现在几个看似无关的上下文中时，对许多数学家来说，有一种令人信服的美学必要性来找到根本原因。有一个伟大的传统，致力于这些问题，虽然答案往往是平凡的，有时他们是相当深刻的。

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.