Theory of multi-wavelet and its application

点击翻译按钮获取中文摘要

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.