Mathematical Sciences: Analysis of Two-Scale Dilation Equations

本文研究的是描述标度函数的双尺度行为特征的双尺度膨胀方程。这些方程出现在各种各样的应用中，包括小波，计算机生成曲线和曲面的细分方案，以及分形几何（迭代函数系统- IFS）。首席研究员将开发基于无限矩阵乘积展开的缩放函数表示的技术，该技术可用于建立构造缩放函数的算法，并分析其规律性，可积性和维数。到目前为止，为了保证无穷矩阵乘积的收敛性，我们提出了一个收缩性假设。这个假设比必要的更强，PI正在研究涉及有界半群矩阵的较弱假设，以及由李雅普诺夫指数确定的平均收缩因子。这也导致了关于矩阵有界半群结构的问题。PI还开发了研究自相似点阵的技术，它源于多维正交小波。PI拟研究以下内容：(1)曲面生成：滑动窗法；(2)可积解的存在性；(3)非负系数膨胀方程；(4)自相似格砖；(5) Daubechies和Lagarias的有限性猜想。这项活动在许多领域都有很大的应用潜力，包括压缩算法。＊＊＊

9307601 Wang This research concerns two-scale dilation equations, which describe the two-scale behavior characteristic of scaling functions. These equations arise in a wide variety of applications, including wavelets, sub-division schemes for computer generation of curves and surfaces, and fractal geometry (iterated function systems - IFS). The principal investigator will develop techniques based on representations of the scaling function in terms of an infinite matrix product expansion, which can be used to establish an algorithm for constructing scaling functions, and to analyze their regularity, integrability, and dimensionality. Thus far, in order to ensure convergence of the infinite matrix product,a Contractivity Assumption has been imposed. This Assumption is stronger than necessary, and the PI is investigating weaker assumptions involving bounded semi-groups of matrices, and an average contractivity factor as determined by the Lyapunov exponent. This also leads to questions regarding the structure of bounded semi-groups of matrices. The PI has also developed techniques to study self-similar lattice tilings, which arises from multi-dimensional orthonormal wavelets. The PI intends to study the following: (1) surface generation: sliding window method; (2) existence of integrable solutions; (3) dilation equations with non-negative coefficients; (4) self-similar lattice tilings; (5) the Finiteness Conjecture of Daubechies and Lagarias. This activity had substantial potential for application in many areas, including compression algorithms. ***