Upper Bound Estimation of Fractal Dimensions of Fractional Integral of Continuous Functions

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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]

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, Rn, 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

{ }

U_i is a countable collection of sets of diameter at most δ that cover F, i.e.

1 i i

F⊂



∝₌U with 0<Ui ≤δ for each i, we say that

{ }

Ui is a δ-cover of F.

Suppose that F is a subset of Rn and s is a non-negative number. For any positive number define.

( )

{ }

1

inf s: is a -cover of s

i i

i

F U U F

δ δ

∞ =

 

=  



∑



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Y. S. Liang

28

Write

( )

0 lim

s s

F _δ F

δ →

=

 

( )

s _F

δ

 is called s-dimensional Hausdorff measure of F. Hausdorff dimension is defined as follows:

Definition 1.1 [5] Let F be a subset of _Rn_and_s_{is a non-negative number. Hausdorff dimension of}_F_is

( )

{

( )

}

{

( )

}

dimH F =inf s:s F =0 =sup s:s F = ∞ If s=dimH

( )

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 _Rn_{and let}

( )

N_δ F 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

( )

log

dim lim

log B

N F

F = δ→ ₋ δ _δ (1.1)

and

( )

0

( )

log

dim lim

log B

N F

F δ δ

δ

→

=

− (1.2)

If (1.1) and (1.2) are equal, we refer to the common value as Box dimension of F

( )

0

( )

log

dim lim

log B

N F

F _δ δ

δ

→

=

− (1.3)

Definition 1.3 [6]Let f x

( )

∈C_{[ ]}_0,1 and v>0. For t∈

[ ]

0,1 we call

( )

_{( ) (}

)

1

( )

0 1

d

x v

v

D 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 D−vf x

( )

be Riemann-Liouville integral of f x

( )

of order v. Then, D−vf 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 1

x v

v

D f x x t f t t v

v

−

− ₌ ₋ _{< <}

Γ

∫

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( )

_{( )}

₍

₎

0 1 1

v M v M

D f x x v

v v v

− _≤ _≤ _{< <}

Γ Γ +

( )

v

D− f x is a bounded function on I.

2.2. Fractal Dimensions of Riemann-Liouville Fractional Integral of

f x

( )

Theorem 2.2 Let v

( )

D− f x be Riemann-Liouville integral of f x

( )

of order v. Then,

(

)

(

)

1≤dim_HΓ D−vf I, ≤dimBΓ D−vf I, ≤ −2 v, 0< <v 1

Proof. Let 0< <δ 1 2, and m is the least integer greater than or equal to 1δ . If 0≤a₁< ≤b₁ δ , we have

( )

(

)

( )

(

)

( )

)

(

)

(

)

( )

(

)

( )

1 1 1 1 1 1 1

1 1 ₀ 1 ₀ 1

1 1 1

0

d d

d d .

b v a v

v v

a v v b 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 ,

max

i x i i

M _{∈ −}_ _{δ δ}_ f x

 

= , _{( )}

( )

1 ,

min

i x i i

m _{∈ −}_ _{δ δ}_ f x

 

= M =maxx I∈ f x

( )

If

( )

1

( )

1 0

v v

D− f b −D− f a ≥ , it holds

(

1

)

( )

1

( ) (

1 1 1

) (

1 1

)

(

1 1

)

1 v

v

v v v

v D− f b D− f a  b a M m b a m

Γ + _ − _≤ − − + −

If D−vf b

( )

₁ −D−vf a

( )

₁ <0, it holds

(

1

)

( )

1

( ) (

1 1 1

) (

1 1

)

v

v v

v D− f b D− f a b a M m

Γ + − ≤ − −

We have

( )

1

( )

1

₍

_{) (}

1 1

) (

1 1

)

1 1

v

v v v

D f b D f a b a M m M

v δ

− ₋ − _≤ ₋ ₋ ₊

Γ +

Let 1≤ ≤ −n m 1. If nδ ≤an+₁≤bn+₁≤

(

n+1

)

δ , we have

( )

(

)

(

)

( )

(

)

( )

(

)

(

)

( )

1 1 1 1

1 1 ₀ 1 ₀ 1

1 1 1 1 0 d d d n n

n b v n a v

v v

n n n n

n v v

n 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 1 1 1 1 1 d d . n n n

n a v v

n n

n

n b v

n n a

n b t n a t f t t

n b t f t t

δ δ δ δ δ δ δ + + + + − − + + + − + +   + _ + − − + − _ + + −

∫

If v

(

₁

)

v

(

₁

)

0

n n

D− f b₊ −D− f a₊ ≥ , it holds

(

1

)

( )

1

(

1

) (

1 1

) (

1 1

)

(

1 1

)

1

v

v v v v

n n n n n n n n n

v D− f b+ D− f a+  b+ a+ M + m+ b+ a+ m+

Γ + _ − _≤ − − + −

If D−vf b

(

_n₊₁

)

−D−vf a

(

_n₊₁

)

<0, it holds

(

1

)

( )

1

(

1

) (

1 1

) (

1 1

) ( )

2

v v

n n n n n n

v D− f b+ D− f a+ b+ a+ M + m+

δ

M

Γ + − ≤ − − +

We get

( )

1

(

1

)

₍

_{) (}

1 1

) (

1 1

)

1

2 1

v

v v v

n n n n n n

D f b D f a b a M m M

v δ

− −

+ − + ≤_{Γ +} + − + + − + +

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Y. S. Liang

30

( )

, 1 , 1 1

v

v D f

R − i

δ

i+

δ

≤C

δ

≤ ≤ −i m

If

(

v

)

N_δ D− f is the number of squares of the

δ

mesh that intersects Γ

(

D−vf I,

)

, by Proposition 11.1 of

[1], we have

(

)

1 1

(

)

2

0

2 v , 1

m

v v

D f i

N_δ D f m δ R − iδ i δ Cδ

−

− − −

=

≤ +

∑

_ + _≤

From (1.2) of Definition 1.2, we know

(

)

0

(

)

log

dim , lim 2 , 0 1

log

v v

B

N D f

D f I δ δ v v

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.

References

[1] Liang, Y.S. (2010) Box Dimension of Riemann-Liouville Fractional Integral of Continuous Function of Bounded Vari- ation. Nonlinear Analysis Series A: Theory, Method and Applications, 72, 2758-2761.

[2] Yao, K. and Liang, Y.S. (2010) The Upper Bound of Box Dimension of the Weyl-Marchaud Derivative of Self-Affine 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.

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