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information Theory under the Method of Functional Analysis

泛函分析方法下的信息论

基本信息

批准号：
03640185
负责人：
UMEGAKI Hisaharu
金额：
$ 1.15万
依托单位：
Meisei University (1992-1993)Tokyo University of Science (1991)
依托单位国家：
日本
项目类别：
Grant-in-Aid for General Scientific Research (C)
财政年份：
1991
资助国家：
日本
起止时间：
1991 至 1993
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-03640185/
关键词：
Entropy Information theory Functional analysis Fourier transform Measurement theory Signal analysis Sampling functions Sufficiency of statistics エントロピ-Fourier解析標本函数係 Gibbs状態

项目摘要

Since the middle of the 20th-century, the field of mathematics is rapidly developing, in particular, the applications of mathmatics to information theory are remarkable. The fundamental part of information theory is so-called informational-mathematical science(IMS).In the present project, using the several methods of functional analysis, the IMS is developed as the followings.(1) Theoretical development of the concept of entropy as information amount.(2) Using the results obtained in (1), IMS-treatments of uncertainty relation in quantum mechanics are studied and we get the uncertainty principle as an entropy formula combined with Fourier transform.(3) In order to construct the concept of information sourse in IMS-form, we set as an alphabet A as a collection of all elements which belong to an information sourse. In the bilateral infinite product A^Z of A, several mathematical structures are defined, and A^Z is totally disconnected compact metric space. This space A^Z is useful as a structure of information source.(4) A fundamental part in the signal analysis is development of sampling expansion theory. For this, we study the sampling function and the orthogonal sequence, in a functional Hilbert space, generated by the sampling function. Those produce various matters among them. We explaine there two main cases. One of them is such that a Hilbert space, of signal functions with bandlimited frequency, which is just equal to a reproducing kernel Hilbert space.Another is the following : the spectral measure of Schrodinger operator is just constructed by sampling functions. These facts are going to clear up the structure of signal functions with finiteenergy.

20世纪中叶以来，数学领域迅速发展，特别是数学在信息论中的应用令人瞩目。信息论的基础部分是所谓的信息数学科学（IMS）。在本项目中，利用泛函分析的几种方法，IMS发展如下：（1）作为信息量的熵概念的理论发展。（2）利用（1）中获得的结果，研究量子力学中不确定性关系的IMS处理，并结合熵公式得到不确定性原理 (3)为了构建IMS形式的信息源概念，我们将属于信息源的所有元素的集合设置为字母表A。在A的双边无穷乘积A^Z中，定义了几种数学结构，A^Z是完全不连通的紧度量空间。这个空间A^Z作为信息源的结构是有用的。(4)信号分析的一个基本部分是采样展开理论的发展。为此，我们在函数希尔伯特空间中研究采样函数和由采样函数生成的正交序列。它们之间产生各种问题。我们解释了两种主要情况。其中之一是具有带限频率的信号函数的希尔伯特空间，其恰好等于再现核希尔伯特空间。另一个是：薛定谔算子的谱测度仅由采样函数构造。这些事实将阐明有限能量信号函数的结构。