2001 Fiscal Year Final Research Report Summary
Algorithmic formulation of wavelet analysis
| Project/Area Number |
12640109
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | NIIGATA UNIVERSITY |
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Principal Investigator |
AKASHI Shigeo Faculty of Science, Niigata University, Professor, 理学部, 教授 (30202518)
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| 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)
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| Project Period (FY) |
2000 – 2001
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| Keywords | wavelet analysis / Furier analysis / E-entropy / Hilbert's 13th problem / Kolmogorov-Arnold representation / superposition representation / the problem of nomographs |
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| 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.
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