RELATIONSHIP OF UPPER BOX DIMENSION BETWEEN CONTINUOUS FRACTAL FUNCTIONS AND THEIR RIEMANN-LIOUVILLE FRACTIONAL INTEGRAL

被引：8
作者：
Xiao, Wei [1 ]
机构：
[1] Nanjing Univ Sci & Technol, Sch Math & Stat, Nanjing 210094, Peoples R China
来源：
FRACTALS-COMPLEX GEOMETRY PATTERNS AND SCALING IN NATURE AND SOCIETY | 2021年 / 29卷 / 08期
关键词：
Riemann-Liouville Fractional Integral; Continuous Fractal Function; Upper Box Dimension; Fractal Dimension; BESICOVITCH FUNCTIONS; DERIVATIVES; CALCULUS;
D O I：
10.1142/S0218348X21502649
中图分类号：
O1 [数学];
学科分类号：
0701 ; 070101 ;
摘要：
This paper considers the relationship of box dimension between a continuous fractal function and its Riemann-Liouville fractional integral. For an arbitrary fractal function f(x) it is proved that the upper box dimension of the graph of Riemann-Liouville fractional integral D(-nu)f(x) does not exceed the upper box dimension of f(x), i.e. dim over bar(B)Upsilon(D-nu f,I) <= dim over bar(B) Upsilon(f,I). This estimate shows that nu-order Riemann-Liouville fractional integral D(-nu)f(x) does not increase the fractal dimension of the integrand f(x), which means that Riemann-Liouville fractional integration does not decrease the smoothness at least that is obvious known result for classic integration. Our result partly answers fractal calculus conjecture in [F. B. Tatom, The relationship between fractional calculus and fractals, Fractals 2 (1995) 217-229] and [Y. S. Liang and W. Y. Su, Riemann-Liouville fractional calculus of one-dimensional continuous functions, Sci. Sin. Math. 4 (2016) 423-438].
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页数：6
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