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Algorithmic formulation of wavelet analysis

小波分析的算法公式

基本信息

批准号：
12640109
负责人：
AKASHI Shigeo
金额：
$ 2.18万
依托单位：
NIIGATA UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2001
项目状态：
已结题

项目摘要

The contents of researches published during two years from 2000 till 2001 can be classified into two parts, which are stated as follows :(1). The solution to an open problem related to Hilbert's 13th problem. The 13th problem formulated by Hilbert, which asks if any continuous functions of several variables can be represented as superposition representation constructed from several continuous functions of fewer variables, was solved by Kolmogorov and Arnold after fifty years. Actually, the analytic function theoretic problem asking if any analytic functions of several variables can be represented as superposition representation constructed from several analytic functions of fewer variables has remained to be solved. The representative of this research project has succeeded in giving the solution to this problem which is based on entropy theoretic methods.(2). Application of the result (1) to the theory of data compression. It is known that the 13th problem is closely related to the problem of nomographs, which belongs to the theory of data compression. Exactly speaking, the set of all analytic functions of several variables, which can be represented as superpositions constructed from several analytic functions of fewer variables, is a subset of the first category. This solution implies that few analytic functions can be represented as superpositions constructed from several functions of fewer variables. Moreover, both this result and the wavelet analysis are also applied to the solution to the problem of generalized nomographs.

2000年至2001年两年间发表的研究内容可分为两个部分，具体如下：(1)一个公开问题的解决与希尔伯特的第13个问题有关。Hilbert提出的第13个问题，即是否有多个变量的连续函数可以表示为由几个变量较少的连续函数构成的叠加表示，五十年后，Kolmogorov和Arnold解决了这个问题。实际上，关于多元解析函数是否可以表示为由几个变量较少的解析函数构成的叠加表示的解析函数论问题一直没有得到解决。本研究项目的代表成功地给出了这一问题的基于熵理论的解决方案。结果(1)在数据压缩理论中的应用。众所周知，第13个问题与诺模图问题密切相关，诺模图问题属于数据压缩理论。确切地说，由多个变量的所有解析函数组成的集合是第一个范畴的一个子集，它可以表示为由几个较少变量的解析函数构成的叠加。这个解意味着，很少有解析函数可以表示为由几个变量较少的函数构成的叠加。此外，这一结果和小波分析也被应用于广义诺模图问题的求解。