Upper Bound Estimation of Fractal Dimensions of Fractional Integral of Continuous Functions ()
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 fractional 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,
, the diameter of U is defined as
, i.e. the greatest distance apart of any pair of points in U. If
is a countable collection of sets of diameter at most δ that cover F, i.e.
with
for each i, we say that
is a δ-cover of F.
Suppose that F is a subset of
and s is a non-negative number. For any positive number define
![](//html.scirp.org/file/3-5300815x12.png)
Write
![](//html.scirp.org/file/3-5300815x13.png)
is called s-dimensional Hausdorff measure of F. Hausdorff dimension is defined as follows:
Definition 1.1 [5] Let F be a subset of
and s is a non-negative number. Hausdorff dimension of F is
![](//html.scirp.org/file/3-5300815x16.png)
If
, then
may be zero or infinite, or may satisfy
![](//html.scirp.org/file/3-5300815x19.png)
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
and let
be the smallest number of sets of diameter at most
which can cover F. Lower and upper Box dimensions of F respectively are defined as
(1.1)
and
(1.2)
If (1.1) and (1.2) are equal, we refer to the common value as Box dimension of F
(1.3)
Definition 1.3 [6] Let
and
. For
we call
![]()
Riemann-Liouville integral of
of order v.
2. Riemann-Liouville Fractional Integral of 1-Dimensional Fractal Function
Let
be a 1-dimensional fractal function on I. We will prove that Riemann-Liouville fractional integral of
is bounded on I. Box dimension of Riemann-Liouville fractional integral of
will be estimated.
2.1. Riemann-Liouville Fractional Integral of ![]()
Theorem 2.1 Let
be Riemann-Liouville integral of
of order v. Then,
is bounded.
Proof. Since
is continuous on a closed interval I, there exists a positive constant M such that
![]()
From Definition 1.3, we know
![]()
For any
, it holds
![]()
is a bounded function on I.
2.2. Fractal Dimensions of Riemann-Liouville Fractional Integral of ![]()
Theorem 2.2 Let
be Riemann-Liouville integral of
of order v. Then,
![]()
Proof. Let
, and m is the least integer greater than or equal to
. If
, we have
![]()
For
, let
, ![]()
If
, it holds
![]()
If
, it holds
![]()
We have
![]()
Let
. If
, we have
![]()
If
, it holds
![]()
If
, it holds
![]()
We get
![]()
There exists a positive constant C, such that
![]()
If
is the number of squares of the
mesh that intersects
, by Proposition 11.1 of [1] , we have
![]()
From (1.2) of Definition 1.2, we know
![]()
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.