Applications of analyticoty of functions

函数解析法的应用

基本信息

批准号：
10304009
负责人：
SAITOH Saburou
金额：
$ 20.42万
依托单位：
Gunma University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (A).
财政年份：
1998
资助国家：
日本
起止时间：
1998 至 2000
项目状态：
已结题

项目摘要

(1) About mathematics, we discussed and gave the definitions of mathematics and good results in mathematics. In particular, the essentials of mathematics are stated to be relations. As an example in this sprit, we established a method connecting "analyticity of functions" and "nonlinear transforms" and derived various concrete results.(2) We found a method discussing the existence of the solutions for general linear differential equations with variable coefficients. This method gives also a contructing algorithms of the solutions, when there exist the solutions.(3) For many solutions for linear partial differential equations depending time, we found a general principle representing the solutions by means of their local deta in both space and time. We can get similar results for many elliptic linear partial differential equations. This pleasant result was named as "Principle of Telethoscope"(4) We found a general principle introducing various operators among Hilbert spaces by mens of transforms. In particular, we were able to give a general definition of the fundamental operators "convolution". As a very simple case, we derived a very simple and beautiful convolution inequality which is different from the famous Young inequality in the convolution.(5) In the viewpoint of conformal mappings, we found very nice representation formulas and their error estimates in the representations of analytic functions in terms of their local deta, using the Riemannn mapping function.(6) We continued the research for real inversion formulas of the Laplace transform from three directions ; that is, uniformly convergence formulas, error estimates and conditional stability.

（1）关于数学，我们讨论并给出了数学和数学良好结果的定义。特别是，数学的要点被认为是关系。作为此Sprit的一个例子，我们建立了一种连接“函数分析”和“非线性变换”并得出各种具体结果的方法。（2）我们找到了一种讨论具有可变系数的通用线性微分方程的解决方案存在的方法。当存在解决方案时，该方法还提供了解决方案的算法。（3）对于许多线性部分偏微分方程的解决方案，根据时间和时间，我们发现了一个一般原理，代表解决方案在空间和时间上的局部deta。对于许多椭圆线性偏微分方程，我们可以获得类似的结果。这个令人愉快的结果被称为“远程听筒原理”（4）我们发现了一个一般原则，通过一群变换引入了希尔伯特空间中的各种操作员。特别是，我们能够对基本运营商“卷积”提供一个总体定义。作为一个非常简单的情况，我们得出了一种非常简单和美丽的卷积不平等，与卷积中著名的年轻不平等不平等不同。（5）在保质映射的角度，我们发现了非常好的表示公式及其在其本地deta的分析函数中的误差估计及其在其本地deta的表示中，使用了Riemann映射功能。也就是说，均匀收敛公式，误差估计和条件稳定性。