Mathematical Sciences: Analysis of Multi-Dimensional Dilation Equations

尺度方程构成了一类广泛应用的函数方程，包括小波，计算机生成曲线和曲面的细分方案，迭代函数系统，分形几何的基本工具。这个项目继续对缩放方程（缩放函数）的无穷矩阵乘积展开的解的表示进行数学研究。目前，乘积的收敛性依赖于一个收缩性假设。无穷积的项从有限的矩阵集合中选择，并以一定的确定性方式组合。所有可能的有限积的出现导致了对矩阵有界半群的研究。我们将研究涉及有界半群的较弱假设，并在相应的Lyapunov指数乘积上建立较弱的收缩条件。本课题的第二个目标是利用无穷矩阵乘积展开建立迭代函数系统算法，用于构造尺度函数，并分析其规律性和维数。这项工作将主要集中在最重要的应用所在的多维环境中。小波理论的核心是函数在空间和频率上的同时局部分解。这一目标在数学分析、信号处理、统计和数据压缩等方面的应用比比皆是。在将基本原理与迭代函数系统的原理相结合的过程中，这项工作自然会导致压缩和重建电视和其他通信信号的重要技术。

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.