Project/Area Number  01540155 
Research Category 
GrantinAid for Scientific Research (C).

Research Field 
解析学

Research Institution  Science University of Tokyo 
Principal Investigator 
梅垣 寿春 東京理科大学, 理学部, 教授
UMEGAKI Hisaharu Science University of Tokyo, Faculty of Science, Professor, 理学部, 教授 (00015992)

CoInvestigator(Kenkyūbuntansha) 
明石 重男 相模工業大学, 工学部, 講師 (30202518)
WATANABE Noboru Science University of Tokyo, Department of Information Sciences, Assistant, 理工学部, 助手 (70191781)
OHYA Masanori Science University of Tokyo, Department of Information Sciences, Professor, 理工学部, 教授 (90112896)
NAGAKURA Yasujiro Science University of Tokyo, Faculty of Science, Professor, 理学部, 教授 (60112900)

Project Fiscal Year 
1989 – 1990

Project Status 
Completed(Fiscal Year 1990)

Budget Amount *help 
¥1,600,000 (Direct Cost : ¥1,600,000)
Fiscal Year 1990 : ¥800,000 (Direct Cost : ¥800,000)
Fiscal Year 1989 : ¥800,000 (Direct Cost : ¥800,000)

Keywords  Fourier Transforms / Radon Transforms / CTScanner / Signal Analysis / Sampling Functions / Optical Modulations / Genetic Codes / Entropy / Fouricr変換 / Radon変換 / CTスキャナ / 信号解析 / 標本函数 / 光変調 / 遺伝情報 / エントロピ / Fourier変換 / Fourir変換 / CTスキャナ 
Research Abstract 
During the last ten years, the field of mathematics has been rapidly developed to various branches. In particular, the effect of the applications of mathematics to the field of information sciences are remarkable. These are actually an important extension of mathematics, and are called to be mathematical sciences or mathematical information sciences. In this research, making the most use of the functional analysis method, we have aimed at further new branches. In the following we describe them by the items (1)(5). (1) Development of Fourier analysis by functional analysis. Several fundamental theorems of operator algebras are adapted to constructions of Gelfand representation and Fourier transforms. Under these constructions, all fundamental theorems in harmonic analysis are expressed. (2) Applying the results in (1) to Radon transforms, we developed mathematical analysis of CTScanner, which is one of the most important principle in new medical diagnosis. (3) A fundamental method in signal analysis is sampling expansion theorem, where the main tool is sampling function. We have developed the functional analysis around the mathematical treatments. The formulations were done by von Neumann algebras and spectral theory. By these invesligations, the mathematical theory of signal has been clarified. While, we have adapted Shannon theory of entropy to optical communication theory, analysis of genes, quantum fractal theory and several topics, and discussed the usefulness of entropy. We describe the following results : (4) By using the functional analytic methods for the mutual entropy and the construction of quantum channel, we discussed the efficiency of some modulations and the error probability in optical communication processes. (5) We made the phylogenetic tree by using the techniques of information theory, and discussed the evolution of organisms.
