Multidimensional Wavelets in Non-Isotropic Function Spaces

本课题研究了谐波分析和小波相关的多维小波展开的数学理论。该学科的基本问题之一是如何构建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。由于I.Daubechies、R. Coifman和Y. Meyer以及其他许多人的贡献，这些研究领域取得了重大进展。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。本文将从三个方向研究这种非各向同性小波理论。首先是研究与膨胀膨胀相关的标准函数空间的非各向同性类似物。特别地，研究与非各向同性扩张相关的Calderon-Zygmund算子的小波扩展特性。二是构造具有良好时频局域性的正交小波，用于求解大范围非各向同性膨胀膨胀。第三是确定非各向同性的膨胀膨胀，其中良好定域小波的构造是不可能的。更一般地说，这个建议代表了小波分析的工作，小波分析是谐波分析中的一种强大的技术。该技术已经在信号和图像处理中产生了广泛的应用，如JPEG 2000图像压缩系统。期望对多维非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许多其他贡献。

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.