Mathematical Sciences: Analysis of Multi-Dimensional Dilation Equations

数学科学：多维膨胀方程的分析

• 批准号: 9204514
• 负责人: Marc Berger
• 金额: $ 4.8万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1992
• 资助国家: 美国
• 起止时间: 1992-09-01 至 1995-02-28
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9204514&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Analysis; Multi; Dimensional

Scaling equations form a class of functional equations arising in a wide variety of applications including wavelets, sub-division schemes for computer generation of curves and surfaces and iterated function systems, the fundamental tools of fractal geometry. This project continues mathematical research on the representation of solutions to scaling equations (scaling functions) in terms of infinite matrix product expansions. At the present time, the convergence of the product is dependent on a contractivity assumption. The terms of the infinite product are chosen from a finite set of matrices and combined in a certain deterministic manner. That all possible finite products appear lead to investigations into bounded semi-groups of matrices. Work will be done investigating weaker assumptions involving bounded semi-groups and establishing weaker contractivity conditions on the corresponding products in terms of the Lyapunov exponent. A second goal of this project is that of using infinite matrix product expansions to establish iterated function system algorithms for the construction of scaling functions and to analyze their regularity and dimensionality. The work will concentrate primarily in the multi-dimensional setting where the most important applications lie. At the heart of wavelet theory is the simultaneous local decomposition of functions in space and frequency. Applications of this goal abound in mathematical analysis, signal processing, statistics and data compression. In combining the underlying principles with that of iterated function systems, this work leads naturally to important techniques for compression and reconstruction of television and other communication signals.

标度方程构成一类函数方程 在包括小波的各种应用中出现， 计算机生成曲线的细分方案， 曲面和迭代函数系统， 分形几何 这个项目继续数学研究 关于尺度方程解的表示（尺度 函数）在无限矩阵乘积展开方面。 在 目前，产品的收敛取决于 收缩性假设 无穷乘积的项 是从一组有限的矩阵中选择的，并组合成一个 某种确定性的方式。 所有可能的有限乘积 似乎导致调查有界半群 矩阵 工作将做调查较弱的假设 涉及有界半群并建立较弱的 相应乘积的收缩性条件 李雅普诺夫指数 该项目的第二个目标是， 使用无限矩阵乘积展开来建立迭代 函数系统的构建算法 函数，并分析其规律性和维数。 这项工作将主要集中在多维 确定最重要的应用所在。 小波理论的核心是同时局部 函数在空间和频率上的分解。 应用 数学分析，信号处理， 统计和数据压缩。 在结合基础的 原则与迭代函数系统，这项工作 自然地导致重要的压缩技术， 电视和其他通信信号的重建。