| Project/Area Number |
15540182
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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 |
Basic analysis
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| Research Institution | The University of the Air |
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Principal Investigator |
KUMAHARA Keisaku The University of the Air, Faculty of Liberal Arts, Professor, 教養学部, 教授 (60029486)
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| Co-Investigator(Kenkyū-buntansha) |
KOIZUMI Shin Onomichi University, Faculty of Economy, Management and Information Sciences, Associate Professor, 経済情報学部, 助教授 (90205310)
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| Project Period (FY) |
2003 – 2005
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| Project Status |
Completed (Fiscal Year 2005)
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| Budget Amount *help |
¥2,600,000 (Direct Cost: ¥2,600,000)
Fiscal Year 2005: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2004: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 2003: ¥1,100,000 (Direct Cost: ¥1,100,000)
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| Keywords | Fourier transform / homogeneous space / sampling theorem / Radon transform / reconstruction theorem / Riemannian symmetric space / Fourier-Jacobi series / complex semisimple Lie group / フーリエ・ヤコビ変換 / ペセンソンのサンプリング公式 / 不確定性原理 / 対称空間 / リー群 / ユニタリ群 / リーマン対称空関 / フーリエ・ヘルガソン変換 / 実双曲空間 / ハリシュ・チャンドラのc関数 |
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| Research Abstract |
We studied the properties of the Fourier transforms on homogeneous spaces of Lie groups. Especially we focused our attention on the sampling theorem of Shannon and its generalizations. The harmonic analysis on homogeneous spaces has developed as extension of the classical Fourier analysis on a real number line or a plane. The Plancherel theorem and the Paley-Wiener theorem have built the important flow in it. These theorems characterize the images by the Fourier transform of some function spaces. In the case of the real line, the Paley-Wiener theorem can be considered from its self-duality to be the theorem by which functions with Fourier transform image of compact support, that is, band-limited functions, are characterized. Such a function is restriction to the real axis of an exponential type entire function. It is the Shannon sampling theorem that the values in all points of such function are with the values at the discrete points on a real axis. This theorem has played the decisive role in communication theory. In this research, we mainly dealt with the problem of the econstruction of the function by the sampling of the Radon conversion on symmetrical spaces.
We obtained the following results :
(1)A sampling formula on the sphere. (2)A sampling formula for Radon transform on the real hyperbolic space, (3)A reconstruction formula on real hyperbolic space using discrete samples of Radon transform, (4)A sampling formula on the complex sphere, (5)A sampling formula for Radon transform on complex hyperbolic space, (6)A reconstruction formula on complex hyperbolic spaces using discrete samples of Radon transform, (7)A reconstruction formula on Riemannian symmetric spaces by using discrete samples of Radon transform, (8)A sampling formula for the Fourier-Jacobi series, (9)A sampling formula on complex semisimple Lie groups.
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