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的定义和数学方面的良好成果。特别是，数学的本质被认为是关系。作为这一精神的一个例子，我们建立了一种连接“函数解析性”和“非线性变换”的方法，并得出了各种具体结果。(2)我们找到了讨论变系数一般线性微分方程解的存在性的方法。当存在解时，该方法还给出了解的构造算法。（3）对于依赖于时间的线性偏微分方程的许多解，我们发现了通过它们在空间和时间上的局部数据来表示解的一般原理。对于许多椭圆线性偏微分方程我们可以得到类似的结果。这个令人愉快的结果被命名为“望远镜原理”(4) 我们发现了一个通过变换在希尔伯特空间中引入各种算子的一般原理。特别是，我们能够给出基本运算符“卷积”的一般定义。作为一个非常简单的情况，我们推导出了一个非常简单而漂亮的卷积不等式，它与卷积中著名的杨氏不等式不同。（5）从共形映射的角度来看，我们发现了非常好的表示公式及其表示中的误差估计(6)从三个方向继续研究拉普拉斯变换的实数反演公式；即一致收敛公式、误差估计和条件稳定性。