1993 Fiscal Year Final Research Report Summary
information Theory under the Method of Functional Analysis
| Project/Area Number |
03640185
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| Research Category |
Grant-in-Aid for General Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Research Field |
解析学
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| Research Institution | Meisei University (1992-1993)
Tokyo University of Science (1991) |
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Principal Investigator |
UMEGAKI Hisaharu Meisei University, Department of Management-Information Sciences, 情報学部, 教授 (00015992)
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| Co-Investigator(Kenkyū-buntansha) |
TSUKADA Makoto Toho Unuversity, Department of Information Sciences, Associate Professor, 理学部, 助教授 (10120198)
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| Project Period (FY) |
1991 – 1993
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| Keywords | Entropy / Information theory / Functional analysis / Fourier transform / Measurement theory / Signal analysis / Sampling functions / Sufficiency of statistics |
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| Research Abstract |
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.
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