Mathematical Sciences: Orthogonal Multiwavelet Constructions

该奖项支持的Hardin DMS-9500905工作重点是小波分析的数学发展及其应用。这是调和分析的一个分支，与许多科学领域的应用密切相关。小波分析的基本目标是(I)确定其整数平移和二进展开形成平方可积函数的正交(或双正交)基的函数(Ii)确保基元同时定位于时间和频率，以及(Iii)它们具有问题所规定的必要的光滑性。通常情况下，小波也应该有紧凑的支持。这个项目将集中于构造二维和更高维紧支撑的正交小波，新的滤波器组结构，以及在图像处理和微分方程组数值方法中的应用。随着物理学家、计算机科学家和与信号处理和数据压缩有关的工程师的几项重要发现的出现，众所周知的小波理论实际上是在过去十年里戏剧性地发展的一系列思想。它演变为许多现有技术的综合，形成了一个框架，为改进应用和挑战数学思想提供了可能性，这将需要数年时间才能达到可能被认为是成熟的阶段。许多科学领域现在都在调整小波结构以适应目前正在研究的重要问题。其中包括统计学家在大数据集中寻找模式、压缩指纹数据、图像边缘重建以及生成具有指定特征的分形集。

Hardin DMS-9500905 Work supported by this award focuses on the mathematical development of wavelet analysis and its applications. This is a branch of harmonic analysis closely allied with applications to many areas of science. The underlying goals of wavelet analysis are (i) to determine functions whose integer translates and dyadic dilates form orthogonal (or biorthogonal) bases for square-integrable functions (ii) to ensure that the basis elements localize both time and frequency and (iii) that they have the requisite smoothness dictated by the problem. Normally, wavelets are also expected to have compact support. This project will focus on construction of compactly supported and orthogonal wavelets in two and higher dimensions, new filter-bank constructions, and application to image processing and numerical methods for differential equations. The theory of wavelets, as it has become known, is actually a body of ideas which has developed dramatically over the past decade, following several important discoveries by physicists, computer scientists and engineers concerned with signal processing and data compression. It evolved into a synthesis of many existing techniques into a framework which offers possibilities for improved applications and challenging mathematical ideas which will require years to reach what might be considered a mature stage. Many areas of science are now adapting wavelet constructs to important problems currently under investigation. These included statisticians looking for patterns in large data sets, compression of fingerprint data, edge reconstruction of images and the generation of fractal sets with prescribed characteristics.