An Operator Approach to Problems in Analysis and Probability: Matrix Muckenhoupt Weights, Hankel and Toeplitz Operators, Singular Integrals and the Angle between Past and Future

分析和概率问题的算子方法：矩阵 Muckenhoupt 权重、Hankel 和 Toeplitz 算子、奇异积分以及过去与未来之间的角度

• 批准号: 9622936
• 负责人: Serguei Treil
• 金额: $ 12.13万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-06-01 至 1999-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9622936&HistoricalAwards=false
• 关键词: Operator; Approach; Problems; Analysis; Probability

Abstract Treil/Volberg DMS-9622963 Treil and Volberg will continue their research on bases of classical wavelets in vector weighted spaces. They will consider some classical problems in the theory of Hankel and Toeplitz operators in spaces of analytic functions, singular integral operators. One goal is the generalizations of these problems to multi-parameter (vector) setting. Other problems they will investigate are Sarason's problem about necessary and sufficient condition for a product of two (unbounded) Toeplitz operators to be bounded, description of completely regular multivariate stationary Gaussian processes, two weights inequality for Bergman Projection, and boundedness of a product of two (big) Hankel operators in the Bergman space. Singular integrals is one of the main tools analysis brought to applications (signal processing, market models, prediction theory, etc..). In practice many such problems have a vector (multi-parameter) nature. The proposed research is based on the approach targeting exactly the multi-parameter nature of such problems and overcomes the difficulty stemming from this multi-parameter nature. Among the practical applications of the research are prediction theory for stationary Gaussian processes, wavelets (signal processing). The authors strongly feel that the developed technique can be helpful.

摘要 Treil/Volberg DMS-9622963 Treil 和 Volberg 将继续他们在向量加权空间中的经典小波基础上的研究。 他们将考虑解析函数空间中 Hankel 和 Toeplitz 算子理论、奇异积分算子的一些经典问题。 目标之一是将这些问题推广到多参数（向量）设置。 他们将研究的其他问题包括萨拉森关于两个（无界）托普利茨算子的乘积有界的充分必要条件的问题、完全正则多元平稳高斯过程的描述、伯格曼投影的两个权重不等式以及两个乘积的有界性（大）伯格曼空间中的汉克尔算子。 奇异积分是应用（信号处理、市场模型、预测理论等）分析的主要工具之一。在实践中，许多此类问题具有向量（多参数）性质。 所提出的研究基于精确针对此类问题的多参数性质的方法，并克服了这种多参数性质带来的困难。 该研究的实际应用包括平稳高斯过程的预测理论、小波（信号处理）。作者强烈认为所开发的技术会有所帮助。