Mathematical Sciences: Orthogonal Multiwavelet Constructions

哈丁DMS-9500905 该奖项支持的工作集中在小波分析及其应用的数学发展。 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）确定其整数平移和二进膨胀形成平方可积函数的正交（或双正交）基的函数（ii）以确保基元素在时间和频率上都本地化，以及（iii）它们具有由问题决定的必要平滑度。 通常，小波也被期望具有紧支撑。 本项目将侧重于构造二维和更高维的紧支撑和正交小波，新的滤波器组构造，以及在图像处理和微分方程数值方法中的应用。 小波理论，因为它已经成为众所周知的，实际上是一个机构的想法，已显着发展，在过去十年中，以下几个重要的发现，物理学家，计算机科学家和工程师关心的信号处理和数据压缩。 它演变成一个综合了许多现有的技术到一个框架，提供了可能性，改进的应用程序和挑战数学的想法，这将需要数年时间才能达到什么可能被认为是一个成熟的阶段。 许多科学领域现在正在调整小波结构，以解决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，压缩指纹数据，图像边缘重建和生成具有规定特征的分形集。

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.