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information Theory under the Method of Functional Analysis

Research Project

Project/Area Number 03640185
Research Category

Grant-in-Aid for General Scientific Research (C)

Allocation TypeSingle-year Grants
Research Field 解析学
Research InstitutionMeisei University (1992-1993)
Tokyo University of Science (1991)

Principal Investigator

UMEGAKI Hisaharu  Meisei University, Department of Management-Information Sciences, 情報学部, 教授 (00015992)

Co-Investigator(Kenkyū-buntansha) TSUKADA Makoto  Toho Unuversity, Department of Information Sciences, Associate Professor, 理学部, 助教授 (10120198)
渡邊 昇  東京理科大学, 理工学部, 助手 (70191781)
佐藤 元  東京理科大学, 理学部, 助教授 (00162462)
永倉 安次郎  東京理科大学, 理学部, 教授 (60112900)
Project Period (FY) 1991 – 1993
Project Status Completed (Fiscal Year 1993)
Budget Amount *help
¥1,800,000 (Direct Cost: ¥1,800,000)
Fiscal Year 1993: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 1992: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 1991: ¥700,000 (Direct Cost: ¥700,000)
KeywordsEntropy / Information theory / Functional analysis / Fourier transform / Measurement theory / Signal analysis / Sampling functions / Sufficiency of statistics / エントロピ- / Fourier解析 / 標本函数係 / Gibbs状態
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.

Report

(4 results)
  • 1993 Annual Research Report   Final Research Report Summary
  • 1992 Annual Research Report
  • 1991 Annual Research Report
  • Research Products

    (38 results)

All Other

All Publications (38 results)

  • [Publications] 梅垣壽春: "A spectral property of one parameter family of sampling functions---from signal analysis to functional analysis" QP-PQ,Quantum Probability & Related topics. 7. 453-463 (1992)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] 梅垣壽春: "情報エントロピーと作用素解析" 数理科学(情報科学とエントロピー特集). 34-41 (1992)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] 梅垣壽春: "不確定性とエントロピー定式--Shannon情報量の発展--" 明星大学研究紀要,情報学部. 63-81 (1993)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] 塚田真: "Operator algebraic method for sufficiency of statistics" Research Report on Information Sciences,Toho University. 1-10 (1992)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] 塚田真: "Gibbs states and simulated annealing" Research Report on Information Sciences,Toho University. 1-12 (1993)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] 梅垣壽春: "Proceedings of the Thirteenth Symposium on Applied Functional Analysis---Information Theory and Related Topics---" 応用函数解析研究会, 117 (1991)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] 梅垣壽春: "情報数理の基礎---関数解析的展開---" サイエンス社, 227 (1993)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] H.Umegaki: "A spectral property of one parameter family of sampling functions ---from signal analysis to functional analysis" QP-PQ, Quantum Probability & Related Topics. 453-464 (1991)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] H.Umegaki: "Information entropy and operator analysis" Mathematical Science. 314-341 (1992)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] H.Umegaki: "Uncertainty and entropy formula ---Developments of Shannon information---" Bull.Meisei Univ.Vol.1. 63-83 (1993)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] H.Umegaki: "Proceeding of the Thirteenth Symposium on Applied Functional Analysis ---Information Theory and Related Topics---" Tokyo Institute of Tech.(1991)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] H.Umegaki: "Information mathematics ---under the methods of Functional Analysis---" Science Book company Tokyo. (1993)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] H.Umegaki: "Integral transforms, Mathemarical Information Sciences (MIS, say)" Asakura Bool Company. in the press.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] H.Umegaki: "Message spaces, MIS" Asakura Book Company. in the press.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] M.Tsukada: "Operator algebraic method for sufficiency of statistis" Research Reports on Information Sciences TOHO-IS 1992-3. 1-10 (1992)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] M.Tsukada: "Gibbs states and simulated annealing" Research Reports on Information Sciences TOHO-IS 1993-3. 1-12 (1993)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] M.Tsukada: "Annealing, MIS" MIS, Asakura Book Company. in the press.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1993 Final Research Report Summary
  • [Publications] 梅垣壽春: "A spectral property of one parameter family of sampling functions ---from signal analysis to functional analysis" QP-PQ,Quantum Probability & Related topics. 7. 453-463 (1992)

    • Related Report
      1993 Annual Research Report
  • [Publications] 梅垣壽春: "情報エントロピーと作用素解析" 数理科学(情報科学とエントロピー特集). 34-41 (1992)

    • Related Report
      1993 Annual Research Report
  • [Publications] 梅垣壽春: "不確定性とエントロピー定式--Shannon情報量の発展--" 明星大学研究紀要,情報科学部. 63-81 (1993)

    • Related Report
      1993 Annual Research Report
  • [Publications] 塚田 真: "Operator algebraic method for sufficiency of statistics" Research Reports on Information Sciences,Toho University. 1-10 (1992)

    • Related Report
      1993 Annual Research Report
  • [Publications] 塚田 真: "Gibbs states and simulated annealing" Research Reports on Information Sciences,Toho University. 1-12 (1993)

    • Related Report
      1993 Annual Research Report
  • [Publications] 梅垣壽春: "Proceedings of the Thirteenth Symposium on Applied Fumctional Analysis---Information Theory and Related Topics---" 応用函数解析研究会, 117 (1991)

    • Related Report
      1993 Annual Research Report
  • [Publications] 梅垣壽春: "情報数理の基礎---関数解析的展開---" サイエンス社, 227 (1993)

    • Related Report
      1993 Annual Research Report
  • [Publications] 梅垣 壽春: "A Spectral Property of One Parameter family of Sampling Functions---from Slgnal Anaysis to Functional Analysis" QP-PQ,Quantum Probability & Related Topics. 7. 453-463 (1992)

    • Related Report
      1992 Annual Research Report
  • [Publications] 梅垣 壽春: "Uncertainty Relation and Entropy Formula" Pagamon Press(in Press.).

    • Related Report
      1992 Annual Research Report
  • [Publications] 梅垣 壽春: "情報エントロピーと作用素解析" 数理科学(エントロピー特集). 34-41 (1992)

    • Related Report
      1992 Annual Research Report
  • [Publications] 梅垣 壽春: "不確定性とエントロピー定式 ---- Shannon 情報量の発展 ----" 明星大学情報学部紀要.

    • Related Report
      1992 Annual Research Report
  • [Publications] 塚田 真: "Operator Algebraic Method for Sufficiency of Statistics" Research Reports Information Sciences Toho University. 4. 1-10 (1992)

    • Related Report
      1992 Annual Research Report
  • [Publications] 梅垣 壽春・塚田 真: "情報科学辞典(分担執筆)" 朝倉書店,

    • Related Report
      1992 Annual Research Report
  • [Publications] 梅垣 壽春: "情報数理の基礎 ---- 関数解析的展開 ----" サイエンス社,

    • Related Report
      1992 Annual Research Report
  • [Publications] 梅垣 壽春: "A Spectral Property of One Parameter family of Sampling Functionsーーーfrom Signal Anaysis to Functional Analysis" QPーPQ,Quantum Probability & Related Topics. VI. 453-463 (1991)

    • Related Report
      1991 Annual Research Report
  • [Publications] 梅垣 壽春: "Uncertainty Relation and Entropy Formula" Pagamon Press.

    • Related Report
      1991 Annual Research Report
  • [Publications] 梅垣 壽春: "情報エントロピ-と作用素解析" 数理科学(情報科学とエントロピ-特集).

    • Related Report
      1991 Annual Research Report
  • [Publications] 塚田 真: "Gibbs States and Simulated Annealing" Resarch Reports on Information Sciences,Toho Univ.

    • Related Report
      1991 Annual Research Report
  • [Publications] 梅垣 壽春(編書): "Proceedings of the Thirteenth Symposium on Applied Functional AnalysisーーーInformation Theory and related Topicsーーー" 応用函数解析研究会, 118 (1991)

    • Related Report
      1991 Annual Research Report
  • [Publications] 梅垣 壽春: "情報科学辞典(分担執肇)" 朝倉書店,

    • Related Report
      1991 Annual Research Report
  • [Publications] 塚田 真: "情報科学辞典(分担執肇)" 朝倉書店,

    • Related Report
      1991 Annual Research Report

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Published: 1991-04-01   Modified: 2016-04-21  

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