Mathematical Sciences: Analysis of Two-Scale Dilation Equations

9307601王本研究涉及描述尺度函数的双尺度行为特征的双尺度膨胀方程。这些方程出现在广泛的应用中，包括小波，用于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)。主要研究人员将开发基于无限矩阵乘积展开形式的尺度函数表示的技术，该技术可用于建立构造尺度函数的算法，并分析其正则性、可积性和维度。到目前为止，为了确保无穷矩阵乘积的收敛，施加了一个对比性假设。这一假设比必要的更强，PI正在调查涉及有界半群矩阵的较弱假设，以及由Lyapunov指数确定的平均收缩系数。这也引出了关于矩阵的有界半群的结构的问题。PI还开发了研究自相似晶格平铺的技术，这种格子平铺源于多维正交子波。PI主要研究如下问题：(1)曲面生成：滑动窗方法；(2)可积解的存在性；(3)具有非负系数的膨胀方程；(4)自相似格子拼接；(5)Daubechies和Lagarias的有限猜想。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. ***