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Fractional Calculus of Fractal Interpolation Function on
The paper researches the continuity of fractal interpolation function’s fractional order integral on and judges whether fractional order integral of fractal interpolation function is still a fractal interpolation function on or not. Relevant theorems of iterated function system and Riemann-Liouville fractional order calculus are used to prove the above researched content. The conclusion indicates that fractional order integral of fractal interpolation function is a continuous function on and fractional order integral of fractal interpolation is still a fractal interpolation function on the interval .
Fractal geometry is a subject in which very irregular and complex phenomena and pictures in nature are researched. From the process of fractal development, there have been some effective methods used in studying fractals so far. For example, Mandelbrot [1–5] applied concept of fractal dimension to describe the roughness of fractal curves and fractal surfaces. Barnsley [6, 7] and Massopust [8, 9] proposed that fractal interpolation curve (refer to Figure 1) and fractal interpolation surface (refer to Figure 2) can be applied in fitting and analyzing the shape of the natural graphs. Feng et al. [10–12] proposed concept and principle of fractal variation and used it in estimating the Minkowski dimension of fractal surface and describing the roughness of fractal surface. Li and Wu  applied wavelet analysis in researching fractal geometry. Ran and Tan  and Mark  discussed the relationship between Fourier analysis and wavelets analysis. Generally, researchers always attempt to research fractals through classical integer order calculus. However, it is very rare that fractals are researched through fractional order calculus. Because classical integer order calculus researches smooth curves and surfaces, it almost cannot be applied in analyzing and dealing with fractal problems, while fractional calculus is regarded as an important and effective tool applied in researching fractal interpolation function.
In order to discuss property of fractal interpolation function’s fractional order integral, the following content is discussed that the continuity of fractal interpolation function's fractional integral on and judge whether fractional integral of fractal interpolation function is still a fractal interpolation function on or not. So iterated function system, concepts, and theorems about Riemann-Liouville fractional order integral are used to prove the above problems. The results indicate that fractional order integral of self-affine transformation’s fractal interpolation function is continuous on and it is still a fractal interpolation function on .
2. Main Concepts and Lemmas
Definition 1 (see ). Let , and function is a continuous function on interval and can be integral on any bounded subinterval included in ; then the following formula is called -order Riemann-Liouville fractional integral of .
Definition 2 (see ). A “hyperbolic” iterated function system consists of a complete metric space () together with a finite set of contraction mappings , with respective contractivity mappings factors , for . The abbreviation “IFS” is used for “iterated function system.” The notation for the IFS just announced is and contractivity factor is .
Definition 3 (see ). Let be a set of points, where . An interpolation function corresponding to this set of data is a continuous function such that
The points are called the interpolation points. It is called that the function of interpolates the data and that the graph of passes through the interpolation points.
Lemma 4 (see ). Let be a positive integer greater than 1. Let denote the IFS defined above, associated with the data set Let the vertical scaling factor obey for . Then there is a metric on , equivalent to the Euclidean metric, such that the IFS is hyperbolic with respect to . In particular, there is a unique nonempty compact set , such that
In particular, an IFS of the form is considered, where the mapping is an affine transformation of the special structure
The transformations are constrained by the data according to
and , , , can be solved from (5) and (6) in terms of the data and vertical scaling factor as follows:
Lemma 5 (see ). Suppose is a set of continuous functions which satisfy and . The metric is defined by the following formula:
Then is a complete metric space. Let the real numbers , , , , be defined by (7). Define a mapping by where is the invertible transformation and then is continuous on the interval and is a contraction mapping on , so possesses a unique fixed point in . That is, there exists a function such that
3. The Continuity of Fractal Interpolation Function’s Fractional Order Integral on the Interval
Lemma 6. If is a continuous function on the interval and , then is a continuous function on too.
Proof. Since then where , so is a continuous function on the interval .
Corollary 7. Suppose is a fractal interpolation function on the interval ; then is continuous on too.
Proof. Since fractal interpolation function of affine transformation is a continuous function on , is continuous on . According to Lemma 6, is a continuous function on too.
Corollary 8. Suppose is a fractal interpolation function of affine transformation on the interval ; then can be integrated on .
Proof. From Corollary 7, and since continuous function on finite closed interval is an integrated function, the result of Corollary 8 is right.
4. Judgement Theorem of Fractal Interpolation Function’s Fractional Integral on
Theorem 9. If is a fractal interpolation function of affine transformation on the interval , then is a fractal interpolation function of affine transformation on the interval too.
Proof. , consider the interval , for , . Let iterated function system (IFS) be so So is a fractal interpolation function of affine transformation on the interval [0, ] and its iterated function system (IFS): where ; then , So is a fractal interpolation function on the interval .
There are three acquired results from the above content in this paper. Firstly, the fractional order integral of fractal interpolation function is continuous on the interval . Secondly, the fractional order integral of fractal interpolation function can be integrated on any closed interval . Finally, the fractional order integral of fractal interpolation function is still a fractal interpolation function on the interval .
The fractional order integral’s differentiability of fractal interpolation function and its boxing dimension will be researched in the future.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
This research was supported by the Scientific Research Innovation Foundation for Graduate Students of Jiangsu Province (no. CXZZ13_0686) and the Nanjing Normal University Taizhou College Youth Fund Project (no. Q201234).
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