ON A CLASS OF BESICOVITCHFUNCTIONS TO HAVE EXACT BOX DIMENSION： A NECESSARY AND SUFFICIENT CONDITION On a class of Besicovitch functions to have exact box dimension: A necessary and sufficient condition期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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ON A CLASS OF BESICOVITCHFUNCTIONS TO HAVE EXACT BOX DIMENSION： A NECESSARY AND SUFFICIENT CONDITION

作者单位：

Zhou Songping (Zhejiang Institute of Science and Technology,China Ningbo University,China) He Guolong (Zhejiang Normal University,China)

基金项目：

国家自然科学基金 , 浙江省自然科学基金

摘要：

This paper summarized recent achievements obtained by the authors about the box dimensions of the Besicovitch functions given byB(t) := ∞∑k=1 λs-2k sin(λkt),where 1 ＜ s ＜ 2, λk ＞ 0 tends to infinity as k →∞ and λk satisfies λk 1/λk ≥λ＞ 1. The results show thatlimk→∞ log λk 1/log λk = 1is a necessary and sufficient condition for Graph(B(t)) to have same upper and lower box dimensions.For the fractional Riemann-Liouville differential operator Du and the fractional integral operator D-v,the results show that if λ is sufficiently large, then a necessary and sufficient condition for box dimension of Graph(D-v(B)),0 ＜ v ＜ s - 1, to be s - v and box dimension of Graph(Du(B)),0 ＜ u ＜ 2 - s, to be s uis also lim k→∞logλk 1/log λk = 1.

关键词：

充分必要条件极限函数无穷大算子

On a class of Besicovitch functions to have exact box dimension: A necessary and sufficient condition

Authors:

Email author" target="_blank">Zhou?Songping Email author He?Guolong

Institution:

(1) Faculty of Science, Zhejiang Institute of Science and Technology, China Ningbo University, 310018 Hangzhou, P. R. China;(2) Department of Matehematics, Zhejiang Normal University, 321004 Hangzhou, P. R. China

Abstract:

This paper summarized recent achievements obtained by the authors about the box dimensions of the Besicovitch functions given by

$B(t): = \sum\limits_{k = 1}^\infty {\lambda _k^{s - 2} } \sin (\lambda _k t),$

where 1<s<2, λ_k > 0 tends to infinity as k→∞ and λ_k satisfies λ_k+1/λ_k ≥ λ > 1. The results show that

$\mathop {\lim }\limits_{k \to \infty } \frac{{\log \lambda _{k + 1} }}{{\log \lambda _k }} = 1$

is a necessary and sufficient condition for Graph(B(t)) to have same upper and lower box dimensions. For the fractional Riemann-Liouville differential operator D^u and the fractional integral operator D^−v, the results show that if λ is sufficiently large, then a necessary and sufficient condition for box dimension of Graph(D^−v(B)), 0<v<s−1, to be s−v and box dimension of Graph(D^u(B)), 0<u<2−s, to be s+u is also $\mathop {\lim }\limits_{k \to \infty } \frac{{\log \lambda _{k + 1} }}{{\log \lambda _k }} = 1$ . Research supported by national Natural Science Foundation of China (10141001) and Zhejiang Provincial Natural Science Foundation 9100042 and 1010009).

Keywords:

Weierstrass function Besicovitch function fractal dimension box dimension Hardmard condition BOX DIMENSION EXACT FUNCTIONS CLASS dimension of Graph large fractional integral operator differential operator upper lower sufficient condition results show paper recent achievements dimensions functions

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