Mathematical Sciences: Wavelets Based on Several Scaling Functions and Related Applications

9503282 Wang这个项目将涉及小波理论和应用的研究。需要考虑的问题包括：(1)Battle-Federbush型小波及其相关小波包的构造和表征；(2)从尺度向量生成的正交多分辨率分析中构造多小波；(3)确定矩阵细化方程若干类解的性质；(4)分形多小波在二阶双曲型偏微分方程中的应用；(5)结合多小波对快速小波变换的适应，以研究非线性积分方程。本项目得到了本科院校研究（RUI）活动的支持。RUI活动是基金会努力的一部分，旨在帮助确保科学和工程研究的广泛基础，从而加强本科院校学生的科学和技术培训。RUI计划的具体目标是：(1)支持教师积极参与本科生的高质量研究；(2)加强主要面向本科教学的学术部门的研究环境；(3)促进以本科为主的院校的研究与教育的整合。这个项目将涉及多小波理论的研究。多小波理论是用矩阵膨胀方程代替小波理论中的经典膨胀方程。本文将研究膨胀方程与所产生的小波基的各种特征之间的关系。小波在偏微分方程和积分方程数值解中的应用也将被研究；这些技术将与更经典的算法进行比较。

9503282 Wang This project will involve research in the theory and applications of wavelets. The problems to be considered include: (1) the construction and characterization of wavelets of Battle-Federbush type and their associated wavelet packets; (2) the general construction of multiwavelets from an orthogonal multiresolution analysis generated by scaling vectors; (3) determining properties of certain classes of solutions of matrix refinement equations; (4) applications of fractal multiwavelets to second order hyperbolic partial differential equations; and (5) adaptations of fast wavelet transformations by incorporating multiwavelets in order to study nonlinear integral equations. This project is being supported under the Research in Undergraduate Institutions (RUI) activity. The RUI activity is part of the Foundation's effort to help assure a broad base for science and engineering research, and thereby enhance the scientific and technical training of students in undergraduate institutions. The specific objectives of the RUI program are to: (1) support high quality research by faculty with active involvement of undergraduate students, (2) strengthen the research environment in academic departments that are oriented primarily toward undergraduate instruction, and (3) promote the integration of research and education at predominantly undergraduate institutions. This project will involve research on the theory of multiwavelets. Multiwavelet theory involves replacing the classical dilation equation of wavelet theory by a matrix dilation equation. The relationship of the dilation equation and the various features of the wavelet bases generated will be investigated. The use of wavelets for the numerical solution of partial differential and integral equations will also be studied; these techniques will be compared to more classical algorithms.