Project/Area Number  10304009 
Research Category 
GrantinAid for Scientific Research (A).

Section  一般 
Research Field 
Basic analysis

Research Institution  Gunma University 
Principal Investigator 
SAITOH Saburou Faculty of Engineering, Gunma University, Professor, 工学部, 教授 (10110397)

CoInvestigator(Kenkyūbuntansha) 
YAMAMOTO Masahiro University of Tokyo, Math.Sci. Associated Professor, 大学院・数理科学研究科, 助教授 (50182647)
相川 弘明 島根大学, 総合理工学部, 教授 (20137889)
水田 義弘 広島大学, 総合科学部, 教授 (00093815)
戸田 暢茂 名古屋工業大学, 工学部, 教授 (30004295)
佐藤 宏樹 静岡大学, 理学部, 教授 (40022222)
MORI Seiki Yamagata University. Science Dept. Professor, 理学部, 教授 (80004456)
HAYASHI Nakao Tokyo Science University Science Dept. Professor, 理学部, 教授 (30173016)
SAKAI Makoto Tokyo Metropolitan University Science Dept. Professor, 理学研究科, 教授 (70016129)
SIGA Hirosige Tokyo Institute of Technology. Science Dept. Professor, 理学部, 教授 (10154189)

Project Fiscal Year 
1998 – 2000

Project Status 
Completed(Fiscal Year 2000)

Budget Amount *help 
¥31,900,000 (Direct Cost : ¥31,900,000)
Fiscal Year 2000 : ¥9,900,000 (Direct Cost : ¥9,900,000)
Fiscal Year 1999 : ¥10,600,000 (Direct Cost : ¥10,600,000)
Fiscal Year 1998 : ¥11,400,000 (Direct Cost : ¥11,400,000)

Keywords  Analytic function / reproducing kernel / inverse problem / differential equ / numerical / Riemann surface / Integral transform / Klein group / 解析性 / 逆問題 / 再生核 / 積分変換 / 微分方程式 / 数値解析 / リーセン面 / クライン群 / リーマン面 / フライン群 / 解析関数 / 等角写像 / ポテンシャル / 値分布理論 / 偏微分方程式 
Research Abstract 
(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.
