| Project/Area Number |
14540116
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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 | KOBE UNIVERSITY |
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Principal Investigator |
FUKUYAMA Katusi Kobe University, Faculty of Science, Professor, 理学部, 教授 (60218956)
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| Co-Investigator(Kenkyū-buntansha) |
HIGUCHI Yasunari Kobe University, Faculty of Science, Professor, 理学部, 教授 (60112075)
TAKAYAMA Nobuki Kobe University, Faculty of Science, Professor, 理学部, 教授 (30188099)
TAKANOBU Satoshi Kanazawa University, Faculty of Science, Associate Professor, 理学部, 助教授 (40197124)
NAGASE Noriaki Hirosaki University, Faculty of Science, Associate Professor, 理学部, 助教授 (30228019)
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| Project Period (FY) |
2002 – 2004
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| Project Status |
Completed (Fiscal Year 2004)
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| Budget Amount *help |
¥2,700,000 (Direct Cost: ¥2,700,000)
Fiscal Year 2004: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2003: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2002: ¥1,100,000 (Direct Cost: ¥1,100,000)
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| Keywords | Central limit theorem / Law of the iterated logarithm / Discrepancy / uniformly distributed sequence / random numbers / gap theorem / lacunary series / invariance principles / Riesz-Raikov和 / non-conventional average |
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| Research Abstract |
Let ^2_0 be the class of function f with ∫^1_0 f(x)dx=0,∫^1_0 |f(x)|^2 dx < ∞.
For given sequence {n_k} of increasing integers and given class X⊂ L^2_0 of functions, we set
Ψ[X;{n_k}](t)=<lim sup>___<K→∞><sup>___<f∈X> (Σ^K_<k=1>f(n_kt))/(√<K log log K>)
We investigated the value distribution of this function.
We proved Ψ[{f};{n_k}]【less than or equal】< ||f||_A =Σ{|f^^^(v)|a.e. under Takahashi's gap condtion n_<k+1>/n_k 【greater than or equal】> 1 + c/k^β(c >0,β<1/2). We also proved the uniform version of this result :
Ψ[X;{n_k}]【less than or equal】sup||f||_A a.e.
for X of some conditions.
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