Multidimensional Wavelets in Non-Isotropic Function Spaces

点击翻译按钮获取中文摘要

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.