Mathematical Sciences: Orthogonal Multiwavelet Constructions

该奖项支持的工作重点是小波分析的数学发展及其应用。这是谐波分析的一个分支，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)确定其整数转换和二进扩张形式为平方可积函数的正交（或双正交）基的函数（ii）确保基元素在时间和频率上都是局部化的（iii）它们具有问题所要求的必要的平滑性。通常，小波也被期望有紧凑的支持。本计画将著重于二维及高维紧支撑正交小波的建构、新滤波器组的建构，以及在影像处理和微分方程数值方法上的应用。众所周知，小波理论实际上是在物理学家、计算机科学家和工程师在信号处理和数据压缩方面的几项重要发现之后，在过去十年中迅速发展起来的一个思想体系。它将许多现有技术综合为一个框架，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。许多科学领域正在将小波结构应用于目前正在研究的重要问题。其中包括统计学家在大型数据集中寻找模式，指纹数据的压缩，图像的边缘重建以及具有规定特征的分形集的生成。

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.