Upper Bound Estimation of Fractal Dimensions of Fractional ...Fractal Dimensions of...
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Advances in Pure Mathematics, 2015, 5, 27-30 Published Online January 2015 in SciRes. http://www.scirp.org/journal/apm http://dx.doi.org/10.4236/apm.2015.51003
How to cite this paper: Liang, Y.S. (2015) Upper Bound Estimation of Fractal Dimensions of Fractional Integral of Conti-nuous Functions. Advances in Pure Mathematics, 5, 27-30. http://dx.doi.org/10.4236/apm.2015.51003
Upper Bound Estimation of Fractal Dimensions of Fractional Integral of Continuous Functions Yongshun Liang Faculty of Science, Nanjing University of Science and Technology, Nanjing, China Email: [email protected] Received 1 December 2014; revised 15 December 2014; accepted 1 January 2015
Copyright © 2015 by author and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/
Abstract Fractional integral of continuous functions has been discussed in the present paper. If the order of Riemann-Liouville fractional integral is v, fractal dimension of Riemann-Liouville fractional inte- gral of any continuous functions on a closed interval is no more than 2 − v.
Keywords Box Dimension, Riemann-Liouville Fractional Calculus, Fractal Function
1. Introduction In [1], fractional integral of a continuous function of bounded variation on a closed interval has been proved to still be a continuous function of bounded variation. The upper bound of Box dimension of the Weyl-Marchaud fractional derivative of self-affine curves has given in [2]. Previous discussion about fractal dimensions of frac-tional calculus of certain special functions can be found in [3] [4].
In the present paper, we discuss fractional integral of fractal dimension of any continuous functions on a closed interval.
If U is any non-empty subset of n-dimensional Euclidean space, nR , the diameter of U is defined as { }sup : ,U x y x y U= − ∈ , i.e. the greatest distance apart of any pair of points in U. If { }iU is a countable
collection of sets of diameter at most δ that cover F, i.e. 1 iiF U∝
=⊂
with 0 iU δ< ≤ for each i, we say that { }iU is a δ-cover of F.
Suppose that F is a subset of nR and s is a non-negative number. For any positive number define.
( ) { }1
inf : is a -cover of ssi i
iF U U Fδ δ
∞
=
=
∑
Y. S. Liang
28
Write
( ) ( )0
lims sF Fδδ→=
( )s Fδ is called s-dimensional Hausdorff measure of F. Hausdorff dimension is defined as follows: Definition 1.1 [5] Let F be a subset of nR and s is a non-negative number. Hausdorff dimension of F is
( ) ( ){ } ( ){ }dim inf : 0 sup :s sH F s F s F= = = = ∞
If ( )dimHs F= , then ( )s F may be zero or infinite, or may satisfy
( )0 s F< < ∞
A Borel set satisfying this last condition is called an s-set. Box dimension is given as follows: Definition 1.2 [5] Let F be any non-empty bounded subset of nR and let ( )N Fδ be the smallest number of
sets of diameter at most δ which can cover F. Lower and upper Box dimensions of F respectively are defined as
( ) ( )0
logdim lim
logB
N FF δ
δ δ→=−
(1.1)
and
( ) ( )0
logdim lim
logB
N FF δ
δδ
→=−
(1.2)
If (1.1) and (1.2) are equal, we refer to the common value as Box dimension of F
( ) ( )0
logdim lim
logB
N FF δ
δ δ→=−
(1.3)
Definition 1.3 [6] Let ( ) [ ]0,1f x C∈ and 0v > . For [ ]0,1t∈ we call
( ) ( ) ( ) ( )1
0
1 dx vvD f x x t f t t
v−− = −
Γ ∫
Riemann-Liouville integral of ( )f x of order v.
2. Riemann-Liouville Fractional Integral of 1-Dimensional Fractal Function Let ( )f x be a 1-dimensional fractal function on I. We will prove that Riemann-Liouville fractional integral of ( )f x is bounded on I. Box dimension of Riemann-Liouville fractional integral of ( )f x will be estimated.
2.1. Riemann-Liouville Fractional Integral of ( )f x Theorem 2.1 Let ( )vD f x− be Riemann-Liouville integral of ( )f x of order v. Then, ( )vD f x− is bounded.
Proof. Since ( )f x is continuous on a closed interval I, there exists a positive constant M such that
( ) f x M x I≤ ∀ ∈
From Definition 1.3, we know
( ) ( ) ( ) ( )1
0
1 d 0 1x vvD f x x t f t t v
v−− = − < <
Γ ∫
For any x I∈ , it holds
Y. S. Liang
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( ) ( ) ( ) 0 1
1v vM MD f x x v
v v v− ≤ ≤ < <
Γ Γ +
( )vD f x− is a bounded function on I.
2.2. Fractal Dimensions of Riemann-Liouville Fractional Integral of ( )f x Theorem 2.2 Let ( )vD f x− be Riemann-Liouville integral of ( )f x of order v. Then,
( ) ( )1 dim , dim , 2 , 0 1v vBH D f I D f I v v− −≤ Γ ≤ Γ ≤ − < <
Proof. Let 0 1 2δ< < , and m is the least integer greater than or equal to 1 δ . If 1 10 a b δ≤ < ≤ , we have
( ) ( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( ) ( )
1 1
1 1
1
1 11 1 1 10 0
1 1 11 1 10
d d
d d .
b av vv v
a bv v v
a
v D f b D f a b t f t t a t f t t
b t a t f t t b t f t t
− −− −
− − −
Γ − = − − −
= − − − + −
∫ ∫
∫ ∫
For 1 i m≤ ≤ , let ( ) ( )1 ,maxi x i iM f xδ δ∈ − = , ( ) ( )1 ,mini x i im f xδ δ∈ −
= ( )max x IM f x∈= If
( ) ( )1 1 0v vD f b D f a− −− ≥ , it holds
( ) ( ) ( ) ( ) ( ) ( )1 1 1 1 1 1 11 11 vvv v vv D f b D f a b a M m b a m− − Γ + − ≤ − − + −
If ( ) ( )1 1 0v vD f b D f a− −− < , it holds
( ) ( ) ( ) ( ) ( )1 1 1 1 1 11 vv vv D f b D f a b a M m− −Γ + − ≤ − −
We have
( ) ( ) ( ) ( ) ( )1 1 1 1 1 11
1vv v vD f b D f a b a M m M
vδ− −− ≤ − − +
Γ +
Let 1 1n m≤ ≤ − . If ( )1 1 1n nn a b nδ δ+ +≤ ≤ ≤ + , we have
( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
1 11 11 1 1 10 0
1 11 10
d d
d
n nn b n av vv vn n n n
n v vn n
v D f b D f a n b t f t t n a t f t t
n b t n a t f t t
δ δ
δ
δ δ
δ δ
+ ++ +− −− −+ + + +
− −+ +
Γ − = + − − + −
= + − − + −
∫ ∫
∫( ) ( ) ( )
( ) ( )
1
1
1
1 11 1
11
d
d .
n
n
n
n a v vn nn
n b vnn a
n b t n a t f t t
n b t f t t
δ
δ
δ
δ
δ δ
δ
+
+
+
+ − −+ +
+ −++
+ + − − + −
+ + −
∫
∫
If ( ) ( )1 1 0v vn nD f b D f a− −+ +− ≥ , it holds
( ) ( ) ( ) ( ) ( ) ( )1 1 1 1 1 1 1 1 11 vv v v vn n n n n n n n nv D f b D f a b a M m b a m− −+ + + + + + + + + Γ + − ≤ − − + −
If ( ) ( )1 1 0v vn nD f b D f a− −+ +− < , it holds
( ) ( ) ( ) ( ) ( ) ( )1 1 1 1 1 11 2v vv vn n n n n nv D f b D f a b a M m Mδ− −+ + + + + +Γ + − ≤ − − +
We get
( ) ( ) ( ) ( ) ( )1 1 1 1 1 11 2
1vv v v
n n n n n nD f b D f a b a M m Mv
δ− −+ + + + + +− ≤ − − +
Γ +
There exists a positive constant C, such that
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( ), 1 1 1, vv
D fR i i C i mδ δ δ− + ≤ ≤ ≤ −
If ( )vN D fδ− is the number of squares of the δ mesh that intersects ( ),vD f I−Γ , by Proposition 11.1 of
[1], we have
( ) ( )1
1 2
02 , 1v
mv v
D fi
N D f m R i i Cδ δ δ δ δ−
−− − −
=
≤ + + ≤ ∑
From (1.2) of Definition 1.2, we know
( ) ( )0
logdim , lim 2 , 0 1
log
vv
BN D f
D f I v vδδ
δ
−−
→Γ = ≤ − < <−
With Definition 1.1, we get the conclusion of Theorem 2.2. This is the first time to give estimation of fractal dimensions of fractional integral of any continuous function
on a closed interval.
Acknowledgements Research is supported by NSFA 11201230 and Natural Science Foundation of Jiangsu Province BK2012398.
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Curves. Analysis Theory and Application, 26, 222-227. http://dx.doi.org/10.1007/s10496-010-0222-9 [3] Liang, Y.S. and Su, W.Y. (2011) The Von Koch Curve and Its Fractional Calculus. Acta Mathematic Sinica, Chinese
Series [in Chinese], 54, 1-14. [4] Zhang, Q. and Liang, Y.S. (2012) The Weyl-Marchaud Fractional Derivative of a Type of Self-Affine Functions. Ap-
plied Mathematics and Computation, 218, 8695-8701. http://dx.doi.org/10.1016/j.amc.2012.01.077 [5] Falconer, J. (1990) Fractal Geometry: Mathematical Foundations and Applications. John Wiley Sons Inc., New York. [6] Oldham, K.B. and Spanier, J. (1974) The Fractional Calculus. Academic Press, New York.