Problems in Function Theory and Operator Theory
函数论和算子理论中的问题
基本信息
- 批准号:0700238
- 负责人:
- 金额:$ 13.2万
- 依托单位:
- 依托单位国家:美国
- 项目类别:Standard Grant
- 财政年份:2007
- 资助国家:美国
- 起止时间:2007-08-15 至 2012-07-31
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
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.
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,这一结果以及为获得该结果而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。它将探索如何将已知在理论数学和数值应用中非常有效的特定离散化技术适应于更复杂的几何设置。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。
项目成果
期刊论文数量(0)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
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Richard Rochberg其他文献
Complex hyperbolic geometry and Hilbert spaces with complete Pick kernels
- DOI:
10.1016/j.jfa.2018.08.017 - 发表时间:
2019-03-01 - 期刊:
- 影响因子:
- 作者:
Richard Rochberg - 通讯作者:
Richard Rochberg
Bergman kernel asymptotics for generalized Fock spaces
- DOI:
10.1007/bf02790262 - 发表时间:
2001-12-01 - 期刊:
- 影响因子:0.900
- 作者:
Finbarr Holland;Richard Rochberg - 通讯作者:
Richard Rochberg
Geometry of Five Point Sets in the Complex Ball
- DOI:
10.1007/s11785-024-01502-8 - 发表时间:
2024-03-24 - 期刊:
- 影响因子:0.800
- 作者:
Richard Rochberg - 通讯作者:
Richard Rochberg
The effect of boundary geometry on Hankel operators belonging to the trace ideals of Bergman spaces
- DOI:
10.1007/bf01191818 - 发表时间:
1997-06-01 - 期刊:
- 影响因子:0.900
- 作者:
Steven G. Krantz;Song-Ying Li;Richard Rochberg - 通讯作者:
Richard Rochberg
Richard Rochberg的其他文献
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{{ truncateString('Richard Rochberg', 18)}}的其他基金
Problems in Function Theory and Operator Theory
函数论和算子理论中的问题
- 批准号:
1001488 - 财政年份:2010
- 资助金额:
$ 13.2万 - 项目类别:
Continuing Grant
Problems in Function Theory and Operator Theory
函数论和算子理论中的问题
- 批准号:
0400962 - 财政年份:2004
- 资助金额:
$ 13.2万 - 项目类别:
Continuing Grant
Problems in Function Theory and Operator Theory
函数论和算子理论中的问题
- 批准号:
0070642 - 财政年份:2000
- 资助金额:
$ 13.2万 - 项目类别:
Continuing Grant
Mathematical Sciences/GIG: "Research and Training in Computational Harmonic Analysis"
数学科学/GIG:“计算调和分析的研究和培训”
- 批准号:
9631359 - 财政年份:1996
- 资助金额:
$ 13.2万 - 项目类别:
Continuing Grant
Mathematical Sciences: Research Group in Analysis
数学科学:分析研究小组
- 批准号:
9531967 - 财政年份:1996
- 资助金额:
$ 13.2万 - 项目类别:
Continuing Grant
Mathematical Sciences: Research Group in Harmonic Analysis
数学科学:调和分析研究组
- 批准号:
9302828 - 财政年份:1993
- 资助金额:
$ 13.2万 - 项目类别:
Continuing Grant
Mathematical Sciences: Spaces of Analytic Functions
数学科学:解析函数空间
- 批准号:
8701271 - 财政年份:1987
- 资助金额:
$ 13.2万 - 项目类别:
Continuing Grant
Mathematical Sciences: Spaces of Analytic Functions
数学科学:解析函数空间
- 批准号:
8402191 - 财政年份:1984
- 资助金额:
$ 13.2万 - 项目类别:
Continuing Grant
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