Mathematical Sciences: Coding and Convergence in Ergodic Theory

数学科学：遍历理论中的编码和收敛

• 批准号: 8620132
• 负责人: Karl Petersen
• 金额: $ 4.85万
• 依托单位: University of North Carolina at Chapel Hill
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1987
• 资助国家: 美国
• 起止时间: 1987-07-01 至 1989-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8620132&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Coding; Convergence; Ergodic

Ergodic theory is an active, central area of Modern Analysis. Its origins lie in the theoretical formulation for classical statistical mechanics at the end of the nineteenth century. The modern point of view derives from the profound mathematical theory of recurrence and ergodicity, as developed by Poincare, Birkhoff, von Neumann and other giants earlier in this century. As the subject has evolved and sophistication continually increased, ergodic theory has acquired close relationships with other branches of mathematics, such as dynamical systems, probability theory, functional analysis, number theory, differential topology, and differential geometry, and with applications as far ranging as mathematical physics, information theory, and computer design. Professor Petersen has worked on several aspects of ergodic theory, especially those connected with harmonic anlaysis, probability, and more recently information theory. A major part of his work has concerned maximal functions and almost everywhere convergence; for example, the refinement of the maximal ergodic theorem to an equality, speed of convergence in the ergodic theorem, the ergodic Hilbert transform, and the relationship with spectral theory and harmonic analysis. He has also conducted research in the applications of ergodic theory, specifically symbolic dynamics, to efficient coding of information across noisy channels. Here, Professor Petersen proposes to continue his research into both symbolic dynamics and almost everywhere convergence. In the former subject, he will consider general properties of symbolic dynamical systems on an infinite alphabet as well as a class of examples that relate to the control of signals for magnetic recording. In the latter area, his research will emphasize connections between the ergodic Hilbert transform and the spectral measure of a measure-preserving transformation.

遍历理论是现代分析学的一个活跃的中心领域。它起源于19世纪末经典统计力学的理论表述。现代的观点源于深刻的数学理论的递归和遍历，如庞加莱，伯克霍夫，冯·诺伊曼和其他巨人在本世纪初发展。随着这门学科的发展和复杂性的不断提高，遍历理论与数学的其他分支建立了密切的关系，如动力系统、概率论、泛函分析、数论、微分拓扑和微分几何，并广泛应用于数学物理、信息论、还有计算机设计。彼得森教授从事遍历理论的几个方面的研究，特别是与谐波分析、概率论和最近的信息论有关的方面。他的主要工作是关于极大函数和几乎处处收敛；例如，将极大遍历定理改进为一个等式，遍历定理中的收敛速度，遍历希尔伯特变换，以及与谱理论和谐波分析的关系。他还研究了遍历理论的应用，特别是符号动力学，以有效地跨噪声信道进行信息编码。在这里，彼得森教授建议继续他对符号动力学和几乎无处不在的趋同的研究。在前一个主题中，他将考虑无限字母表上符号动力系统的一般性质，以及一类与磁记录信号控制有关的例子。在后一个领域，他的研究将强调遍历希尔伯特变换与保测度变换的谱测度之间的联系。