Multidimensional Wavelets in Non-Isotropic Function Spaces

这个项目研究了与多维小波展开的数学理论有关的调和分析和小波领域。如何构造具有所需特征的高维小波基，例如具有良好时频特性的小波，是本课题研究的基本问题之一。由于I.Daubechies、R.Coifman和Y.Meyer以及其他许多人的贡献，这些领域的研究取得了重大进展。然而，大多数研究都集中在各向同性理论上，留下了许多涉及非各向同性小波理论的问题没有得到解答。作者将从三个方面研究各向同性小波理论。第一个是研究与扩张相关的标准函数空间的非各向同性类似，特别是研究与非各向同性扩张相关的Calderon-Zygmund算子的小波展开的刻画。二是对大类非各向同性膨胀项构造具有良好时频局部化的正交小波。第三种是识别不可能构造良好局部化小波的各向同性伸缩。更广泛地说，这代表了小波分析的工作，这是一种强大的调和分析技术。这一技术在信号和图像处理中产生了广泛的应用，如JPEG2000图像压缩系统。人们期待着对多维各向同性小波的进一步研究将继续在纯数学和应用数学中产生更多的贡献。

This project investigates the areas of harmonic analysis andwavelets concerning the mathematical theory of multi-dimensional wavelet expansions. One of the fundamentalproblems of the subject is how to construct higher dimensionalwavelet bases with desired characteristics, e.g., wavelets withgood time-frequency properties. These areas of research haveseen significant progress due to the contributions of I.Daubechies, R. Coifman, and Y. Meyer, along with many others.However, the majority of research has been concentrated onisotropic theory, leaving many questions involving non-isotropic wavelet theory unanswered. The proposer willinvestigate this theory of non-isotropic wavelets from threedirections. The first is to study non-isotropic analogues ofthe standard function spaces associated with expansive dilations.In particular, to examine characterization by wavelet expansionsof Calderon-Zygmund operators associated with non-isotropicdilations. The second is to construct orthogonal wavelets withgood time-frequency localization for large classes of non-isotropic expansive dilations. The third is to identify non-isotropic expansive dilations for which the construction ofwell-localized wavelets is impossible.More generally, this proposal represents work on waveletanalysis which is a powerful technique in harmonic analysis.This technique has produced wide-ranging applications to signaland image processing, such as the JPEG 2000 image compressionsystem. It is expected that further research on multi-dimensional non-isotropic wavelets will continue to produce manyother contributions in pure and applied mathematics.