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Algorithmic formulation of wavelet analysis

Research Project

Project/Area Number 12640109
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionNIIGATA UNIVERSITY

Principal Investigator

AKASHI Shigeo  Faculty of Science, Niigata University, Professor, 理学部, 教授 (30202518)

Co-Investigator(Kenkyū-buntansha) SUZUKI Tomonari  Graduate School of Science and Technology, Niigata University, Assistant, 大学院・自然科学研究科, 助手 (00303173)
HATORI Osamu  Graduate School of Science and Technology, Niigata University, Associate Professor, 大学院・自然科学研究科, 助教授 (70156363)
ISOGAI Eiichi  Faculty of Science, Niigata University, Professor, 理学部, 教授 (40108014)
OHYA Masanori  Science University of Tokyo, Department of Science and Technology, Professor, 理工学部, 教授 (90112896)
Project Period (FY) 2000 – 2001
Project Status Completed (Fiscal Year 2001)
Budget Amount *help
¥3,400,000 (Direct Cost: ¥3,400,000)
Fiscal Year 2001: ¥1,600,000 (Direct Cost: ¥1,600,000)
Fiscal Year 2000: ¥1,800,000 (Direct Cost: ¥1,800,000)
Keywordswavelet analysis / Furier analysis / E-entropy / Hilbert's 13th problem / Kolmogorov-Arnold representation / superposition representation / the problem of nomographs / Waveletスペクトル / エントロピー理論 / 計算の複雑さ
Research Abstract

The contents of researches published during two years from 2000 till 2001 can be classified into two parts, which are stated as follows :
(1). The solution to an open problem related to Hilbert's 13th problem. The 13th problem formulated by Hilbert, which asks if any continuous functions of several variables can be represented as superposition representation constructed from several continuous functions of fewer variables, was solved by Kolmogorov and Arnold after fifty years. Actually, the analytic function theoretic problem asking if any analytic functions of several variables can be represented as superposition representation constructed from several analytic functions of fewer variables has remained to be solved. The representative of this research project has succeeded in giving the solution to this problem which is based on entropy theoretic methods.
(2). Application of the result (1) to the theory of data compression. It is known that the 13th problem is closely related to the problem of nomographs, which belongs to the theory of data compression. Exactly speaking, the set of all analytic functions of several variables, which can be represented as superpositions constructed from several analytic functions of fewer variables, is a subset of the first category. This solution implies that few analytic functions can be represented as superpositions constructed from several functions of fewer variables. Moreover, both this result and the wavelet analysis are also applied to the solution to the problem of generalized nomographs.

Report

(3 results)
  • 2001 Annual Research Report   Final Research Report Summary
  • 2000 Annual Research Report
  • Research Products

    (22 results)

All Other

All Publications (22 results)

  • [Publications] Shigeo Akashi: "Embedding of expansive dynamical systems into symbolic dynamical systems"Reports on Mathematical Physics. 46. 11-14 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Shigeo Akashi: "Compactification of the stationary channel space"Journal of Mathematical Analysis and Applications. 250. 320-332 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Shigeo Akashi: "Topological classification of compact nonkinear operators"Nonlinear Analysis. 47. 555-560 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Shigeo Akashi: "Application of ε-entropy to kolmogeror-Arrold Representation"Reports on Mathematical Physis. 48. 19-26 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Shigeo Akashi: "Aversion of Hilbert's 13th problem for analytic functions"to be published in the Bulletin of the London Mathematical Society.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Tomonari Suzuki: "Generalized distance and existence theorems in complete metric spaces"Journal of Mathematical Analysis and Applications. 253. 440-458 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] 明石 重男: "現代数学への展望"横浜図書. 116 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] S. Akashi: "Embedding of expansive dynamical systems into symbolic dynamical systems"Reports on Mathematical Physics. vol. 46. 11-14 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] S. Akashi: "Compactification of the stationary channel space"Journal of Mathematical Analysis and Applications. vol. 250. 320-332 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] S. Akashi: "Topological classification of compact nonlinear operators"Nonlinear Analysis. vol. 47. 555-560 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] S. Akashi: "Application of epsilon-entropy to Kolmogorov-Arnold representation"Reports on Mathematical Physics. vol. 48. 19-26 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] S. Akashi: "A version of Hilbert's 13th problem for analytic functions"the Bulletin of the London Mathematical Society. (to be published).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T. Suzuki: "Generalized distance and existence theorems in complete metric spaces"Journal of Mathematical Analysis and Applications. vol. 253. 440-458 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] S. Akashi: "The prospects pf Modern Mathematics, Yokohama Publishers, 116 pages"Yokohama Publishers. 116 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Shigeo Akashi: "Topological Classification of Compact Nonlinear Mappings"Nonlinear Analysis. 47・1. 555-560 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Shigeo Akashi: "Application of ε-entropy to Kolmogurov-Arnold Representation"Reports on Mathematical Physics. 48・1. 19-26 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Shigeo Akashi: "A Version of Hilbert's 13th Problem for Analytic Functions"The Bulletin of the London Mathematical Society. (to be published)(発売予定).

    • Related Report
      2001 Annual Research Report
  • [Publications] 明石 重男: "エントロピー解析とフラクタル幾何学"Springer Verlag東京(発刊予定).

    • Related Report
      2001 Annual Research Report
  • [Publications] Shigeo Akashi: "Embedding of expansive dynamical systems into symbolic dynamical systems"Reports on Mathematical Physics. 46. 11-14 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Shigeo Akashi: "Compactification of the Stationary channel space"Journal of Mathematical Analysis and Applications. 250. 320-332 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Tomonari Suzuki: "Generalized distance and existence theorems in complete metric spaces"Journal of Mathematical Analysis and Applications. 253. 440-458 (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] 明石重男: "現代数学への展望"横浜図書. 116 (2000)

    • Related Report
      2000 Annual Research Report

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Published: 2000-04-01   Modified: 2016-04-21  

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