Theory of multi-wavelet and its application

本课题的研究目的是将小波理论应用于实际的生物学问题。小波理论开始于八十年代初，在第一阶段，小波理论的构造只有一个小波函数。因此，构造或寻找适合于实际应用的小波函数是小波理论研究的主要内容。然而，该理论的应用已经非常迅速地扩展，有时我们需要两个以上的小波函数来开发函数或信号。因此，我们研究了多小波理论。本项目的第一年，我们研究了将该理论应用于技术中的实际问题的可能性，例如，电信和图像处理。我们有机会见到许多应用数学家，不仅在日本，而且在几个国家，在小波理论中，我们使用时频分析，这在数学上被称为微局部分析，作为一种基本方法。利用时频分析的方法，我们试图将某些函数（分布）描述为图像，并研究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有着密切的联系。在伪微分算子理论中，我们得到了sharp Garding不等式的一个推广形式。在偏微分方程理论中也得到了许多结果，如超线性方程初值问题解的奇异性研究。用小波或多小波来重新考虑这些问题将是一个有趣的问题，并可能使偏微分方程应用于实际问题中成为可能。

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.