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Theory of multi-wavelet and its application

多小波理论及其应用

基本信息

批准号：
09554001
负责人：
NAGASE Michihiro
金额：
$ 2.43万
依托单位：
Osaka University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
1997
资助国家：
日本
起止时间：
1997 至 1999
项目状态：
已结题

项目摘要

The purpose of this research project is to apply wavelet theory to practical problems in thecnology. The theory of wavelet begins in the early eighties, and at the first stage the wavelet thery was constructed only one wavelet functions. So the main concern of the theory was to construct or look for the wavelet function which was appropriate for the applications. However the application of the theory has been extended very rapidly and sometimes we need more than two wavelet functions to develope functions or signals. That is why we investigate the nulti-wavelet theory.First year of this project we investigated the possibility to apply the theory to the practical problem in technology, for example, telecommunication and image processing. We have had a chance to meet many applied mathematicians not only inside of Japan but also in several countriesIn the theory of wavelet we use the time-frequency analysis, which is called mathematically the microlocal analysis, as a fundamental method. Using the time-frequency analysis we tried to describe some functions (distributions) as pictures and investigated the singularity of functions in the pictures.Theory of wavelet is closely related to the theory of partial differential equations or theory of pseudo-differential operators. In the theory of pseudo-differential operators, we get a generalized form of the sharp Garding's inequality. Also we get many result in the theory of partial differential equations like in he investigation of the singularity for the solution of the intial value problem for hyperolic equations. The reconsideration of these problems by using wavelet or multiwavelet will be interesting problems and may give the possibility of application of PDE to the practical problems in thecnology.

本研究项目的目的是将小波理论应用于实际技术问题。小波理论起步于八十年代初，最初的小波理论只构造了一个小波函数。所以理论的主要关注点是构造或寻找适合应用的小波函数。然而，该理论的应用已经得到了迅速的推广，有时我们需要两个以上的小波函数来表示函数或信号。这就是我们研究零小波理论的原因。在这个项目的第一年，我们研究了将理论应用于技术实际问题的可能性，例如电信和图像处理。在小波理论中，我们使用时频分析，在数学上称为微局部分析，作为一种基本方法。本文尝试用时频分析方法将一些函数（分布）描述为图像，并研究了图像中函数的奇异性。小波理论与偏微分方程理论或伪微分算子理论密切相关。在伪微分算子理论中，我们得到了尖锐Garding不等式的广义形式。在偏微分方程理论中也得到了许多结果，如对双曲方程初值问题解的奇异性的研究。用小波或多小波重新考虑这些问题将是一个有趣的问题，并可能为PDE在实际技术问题中的应用提供可能性。