Research ArticleFractional Calculus of FractalInterpolation Function on [0 119887](119887 gt 0)
XueZai Pan12
1 Faculty of Science Jiangsu University Zhenjiang 212013 China2 School of Mathematics Nanjing Normal University Taizhou College Taizhou 225300 China
Correspondence should be addressed to XueZai Pan xzpan1975163com
Received 16 February 2014 Accepted 20 March 2014 Published 10 April 2014
Academic Editor Baojian Hong
Copyright copy 2014 XueZai Pan This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
The paper researches the continuity of fractal interpolation functionrsquos fractional order integral on [0 +infin) and judges whetherfractional order integral of fractal interpolation function is still a fractal interpolation function on [0 119887](119887 gt 0) or not Relevanttheorems of iterated function system and Riemann-Liouville fractional order calculus are used to prove the above researchedcontentThe conclusion indicates that fractional order integral of fractal interpolation function is a continuous function on [0 +infin)
and fractional order integral of fractal interpolation is still a fractal interpolation function on the interval [0 119887]
1 Introduction
Fractal geometry is a subject in which very irregular andcomplex phenomena and pictures in nature are researchedFrom the process of fractal development there have beensome effective methods used in studying fractals so farFor example Mandelbrot [1ndash5] applied concept of fractaldimension to describe the roughness of fractal curves andfractal surfaces Barnsley [6 7] andMassopust [8 9] proposedthat fractal interpolation curve (refer to Figure 1) and fractalinterpolation surface (refer to Figure 2) can be applied infitting and analyzing the shape of the natural graphs Feng etal [10ndash12] proposed concept and principle of fractal variationand used it in estimating the Minkowski dimension of fractalsurface and describing the roughness of fractal surface Liand Wu [13] applied wavelet analysis in researching fractalgeometry Ran and Tan [14] and Mark [15] discussed therelationship between Fourier analysis and wavelets analysisGenerally researchers always attempt to research fractalsthrough classical integer order calculus However it is veryrare that fractals are researched through fractional ordercalculus Because classical integer order calculus researchessmooth curves and surfaces it almost cannot be applied inanalyzing and dealing with fractal problems while fractionalcalculus is regarded as an important and effective tool appliedin researching fractal interpolation function
In order to discuss property of fractal interpolationfunctionrsquos fractional order integral the following contentis discussed that the continuity of fractal interpolationfunctionrsquos fractional integral on [0 +infin) and judge whetherfractional integral of fractal interpolation function is stilla fractal interpolation function on [0 119887](119887 gt 0) or notSo iterated function system concepts and theorems aboutRiemann-Liouville fractional order integral are used to provethe above problemsThe results indicate that fractional orderintegral of self-affine transformationrsquos fractal interpolationfunction is continuous on [0 +infin) and it is still a fractalinterpolation function on [0 119887](119887 gt 0)
2 Main Concepts and Lemmas
Definition 1 (see [16]) Let V gt 0 and function 119891 is acontinuous function on interval (0 +infin) and can be integralon any bounded subinterval included in [0 +infin) then thefollowing formula
119868minusV119891 (119909) =
1
Γ (V)int119909
0
(119909 minus 119905)Vminus1
119891 (119905) 119889119905 (1)
is called V-order Riemann-Liouville fractional integral of 119891
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 640628 5 pageshttpdxdoiorg1011552014640628
2 Abstract and Applied Analysis
Figure 1 Fractal interpolation curve
Figure 2 Fractal interpolation surface
Definition 2 (see [17]) A ldquohyperbolicrdquo iterated functionsystem consists of a complete metric space (119883 119889) togetherwith a finite set of contraction mappings 119908
for 119894 = 1 2 119899The abbreviation ldquoIFSrdquo is used for ldquoiterated function systemrdquoThe notation for the IFS just announced is 119883 119908
119894
119894 =
1 2 119899 and contractivity factor is 119904 = max119904119894
119894 =
1 2 119899
Definition 3 (see [7]) Let (119909119894
119910119894
) isin 1198772
119894 = 0 1 2 119899
be a set of points where 1199090
lt 1199091
lt 1199092
lt sdot sdot sdot lt 119909119899
Aninterpolation function corresponding to this set of data is acontinuous function 119891 [119909
0
119909119899
] rarr 119877 such that
119891 (119909119894
) = 119910119894
119894 = 1 2 119899 (2)
The points (119909119894
119910119894
) are called the interpolation points It iscalled that the function of119891 interpolates the data and that thegraph of 119891 passes through the interpolation points
Lemma 4 (see [17]) Let 119899 be a positive integer greater than1 Let 119877
2
119908119894
119894 = 1 2 119899 denote the IFS defined aboveassociated with the data set
(119909119894
119910119894
) isin 1198772
119894 = 0 1 2 119899 (3)
Let the vertical scaling factor 119889119894
obey 0 le 119889119894
lt 1 for 119894 =
1 2 119899 Then there is a metric 119889 on 1198772 equivalent to the
Euclidean metric such that the IFS is hyperbolic with respectto 119889 In particular there is a unique nonempty compact set119866 sub 119877
2 such that
119866 =
119899
⋃119894=1
119908119894
(119866) (4)
In particular an IFS of the form 1198772
119908119894
119894 = 1 2 119899 isconsidered where the mapping is an affine transformation ofthe special structure
119908119894
(119909
119910) = (
119886119894
0
119888119894
119889119894
)(119909
119910) + (
119890119894
119891119894
) (5)
The transformations are constrained by the data accord-ing to
119908119894
(1199090
1199100
) = (119909119894minus1
119910119894minus1
) 119908119894
(119909119899
119910119899
) = (119909119894
119910119894
) for 119894 = 1 2 119899
(6)
and 119886119894
119890119894
119888119894
119891119894
can be solved from (5) and (6) in terms of thedata and vertical scaling factor 119889
119894
as follows
119886119894
=119909119894
minus 119909119894minus1
119909119899
minus 1199090
119890119894
=119909119899
119909119894minus1
minus 1199090
119909119894
119909119899
minus 1199090
119888119894
=119910119894
minus 119910119894minus1
119909119899
minus 1199090
minus119889119894
(119910119899
minus 1199100
)
119909119899
minus 1199090
119891119894
=119909119899
119910119894minus1
minus 1199090
119910119894
119909119899
minus 1199090
minus119889119894
(119909119899
1199100
minus 1199090
119910119899
)
119909119899
minus 1199090
(7)
Lemma 5 (see [18]) Suppose 119865 is a set of continuous functionswhich satisfy 119891 [119909
0
119909119899
] rarr 119877 and 119891(1199090
) = 1199100
119891(119909119899
) = 119910119899
The metric is defined by the following formula
119889 (119891 119892) = max 1003816100381610038161003816119891 (119909) minus 119892 (119909)1003816100381610038161003816 119909 isin [119909
0
119909119899
] forall119891 119892 isin 119865
(8)
Then (119865 119889) is a complete metric space Let the real numbers119886119894
119888119894
119890119894
119891119894
be defined by (7) Define a mapping 119879 119865 rarr 119865 by
(119879119891) (119909) = 119888119894
119871minus1
119894
(119909) + 119889119894
119891 (119871minus1
119894
(119909)) + 119891119894
119909 isin [119909119894minus1
119909119894
] 119894 = 1 2 119899
(9)
where 119871119894
[1199090
119909119899
] rarr [119909119894minus1
119909119894
] is the invertible transforma-tion
119871119894
(119909) = 119886119894
119909 + 119890119894
119871minus1
119894
(119909) =119909 minus 119890119894
119886119894
119871minus1
119894
(119909119894minus1
) = 1199090
119871minus1
119894
(119909119894
) = 119909119899
(10)
and then 119879119891 is continuous on the interval [119909119894minus1
119909119894
] and 119879 isa contraction mapping on (119865 119889) so 119879 possesses a unique fixedpoint in 119865 That is there exists a function 119891 isin 119865 such that
(119909 minus 119905)Vminus1 1003816100381610038161003816119891 (119905 + Δ119909) minus 119891 (119905)
1003816100381610038161003816 119889119905)
le1
|Γ (V)|(119909
Vminus1119872Δ119909 + 120576int
119909
0
(119909 minus 119905)Vminus1
119889119905)
=1
|Γ (V)|(119909
Vminus1119872Δ119909 +
120576
V119909V) 997888rarr 0
(13)
where 119872 = max119905isin[0Δ119909]
|119891(119905)| so 119868minusV119891(119909) is a continuous
function on the interval [0 +infin)
Corollary 7 Suppose 119891(119909) is a fractal interpolation functionon the interval [0 +infin) then 119868
minusV119891(119909) is continuous on [0 +infin)
too
Proof Since fractal interpolation function of affine trans-formation is a continuous function on [0 +infin) 119891(119909) iscontinuous on [0 +infin) According to Lemma 6 119868minusV119891(119909) is acontinuous function on [0 +infin) too
Corollary 8 Suppose 119891(119909) is a fractal interpolation functionof affine transformation on the interval [119886 119887](0 lt 119886 lt 119887 lt
+infin) then 119868minusV119891(119909) can be integrated on [119886 119887]
Proof From Corollary 7 and since continuous function onfinite closed interval is an integrated function the result ofCorollary 8 is right
4 Judgement Theorem of FractalInterpolation Functionrsquos FractionalIntegral on [0 119887](119887gt0)
Theorem 9 If 119891(119909) is a fractal interpolation function of affinetransformation on the interval [0 119887](119887 gt 0) then 119868
minusV119891(119909) is
a fractal interpolation function of affine transformation on theinterval [0 119887] too
Proof For all 119887 isin (0 +infin) consider the interval [0 119887] for0 lt V lt 1 119909 isin [119909
119894minus1
119909119894
](119909119894
= (119894119899)119887 119894 = 1 2 119899) Letiterated function system (IFS) be
119871119894
(119909) =1
119899119909 +
119894 minus 1
119899119887
119865119894
(119909 119910) = 119888119894
119909 + 119889119894
119910 + 119891119894
(14)
so119871minus1
119894
(119909) = 119899119909 minus (119894 minus 1) 119887
119868minusV119891 (119909) =
1
Γ (V)int119909
0
(119909 minus 119905)Vminus1
119891 (119905) 119889119905
=1
Γ (V)
119896minus1
sum119894=1
int119909119894
119909119894minus1
(119909 minus 119905)Vminus1
119891 (119905) 119889119905
+1
Γ (V)int119909
119909119896minus1
(119909 minus 119905)Vminus1
119891 (119905) 119889119905
=1
Γ (V)
119896minus1
sum119894=1
int119887
0
(119909 minus1
119899119910 minus
119894 minus 1
119899119887)
Vminus1
times 119891 [119871119894
(119910)]1
119899119889119910
(Let 119905 = 119871119894
(119910))
+1
Γ (V)int119871
minus1
119896(119909)
0
(119909 minus1
119899119910 minus
119896 minus 1
119899119887)
Vminus1
times 119891 [119871119896
(119910)]1
119899119889119910
=1
Γ (V) 119899V119896minus1
sum119894=1
int119887
0
[119899119909 minus 119910 minus (119894 minus 1) 119887]Vminus1
times [119888119894
119910 + 119889119894
119891 (119910) + 119891119894
] 119889119910
+1
Γ (V) 119899Vint119871
minus1
119896(119909)
0
[119899119909 minus 119910 minus (119896 minus 1) 119887]Vminus1
times (119888119896
119910 + 119889119896
119891 (119910) + 119891119896
) 119889119910
4 Abstract and Applied Analysis
=1
Γ (V) 119899V119896minus1
sum119894=1
int119887
0
[119871minus1
119894
(119909) minus 119910]Vminus1
times [119888119894
119910 + 119889119894
119891 (119910) + 119891119894
] 119889119910
+119889119896
Γ (V) 119899Vint119871
minus1
119896(119909)
0
[119871minus1
119896
(119909) minus 119910]Vminus1
119891 (119910) 119889119910
+1
Γ (V) 119899Vint119871
minus1
119896(119909)
0
[119871minus1
119896
(119909) minus 119910]Vminus1
times (119888119896
119910 + 119891119896
) 119889119910
=1
Γ (V) 119899V119896minus1
sum119894=1
int119887
0
[119871minus1
119894
(119909) minus 119910]Vminus1
times [119888119894
119910 + 119889119894
119891 (119910) + 119891119894
] 119889119910
+1
Γ (V) 119899Vint119871
minus1
119896(119909)
0
[119871minus1
119896
(119909) minus 119910]Vminus1
times (119888119896
119910 + 119891119896
) 119889119910 +119889119896
119899V119910
=1
Γ (V) 119899V
times
119896minus1
sum119894=1
[119889119894
int119887
0
(119871minus1
119894
(119909) minus 119910)Vminus1
119891 (119910) 119889119910
minus119887119888119894
V(119871minus1
119894
(119909) minus 119887)Vminus
119888119894
V (V + 1)
times [(119871minus1
119894
(119909) minus 119887)V+1
minus (119871minus1
119894
(119909))V+1
]
+ 119887119891119894
]
+119889119896
Γ (V) 119899Vint119871
minus1
119896(119909)
0
(119871minus1
119896
(119909) minus 119910)Vminus1
119891 (119910) 119889119910
+(119871minus1
119896
(119909))V
Γ (V) 119899V[
119888119896
V (V + 1)119871minus1
119896
(119909) +119891119896
V]
(15)
So 119868minusV119891(119909) is a fractal interpolation function of affine trans-
formation on the interval [0 119887] and its iterated functionsystem (IFS)
119871119894
(119909) =1
119899119909 +
119894 minus 1
119899119887 119894 = 1 2 119899
119865119894
(119909 119910)
=1
Γ (V) 119899V
times
119896minus1
sum119894=1
[119889119894
int119887
0
(119871minus1
119894
(119909) minus 119910)Vminus1
119891 (119910) 119889119910
minus119887119888119894
V(119871minus1
119894
(119909) minus 119887)Vminus
119888119894
V (V + 1)
times [(119871minus1
119894
(119909) minus 119887)V+1
minus (119871minus1
119894
(119909))V+1
] + 119887119891119894
]
+119889119896
Γ (V) 119899Vint119871
minus1
119896(119909)
0
(119871minus1
119896
(119909) minus 119910)Vminus1
119891 (119910) 119889119910
+(119871minus1
119896
(119909))V
Γ (V) 119899V[
119888119896
V (V + 1)119871minus1
119896
(119909) +119891119896
V]
(16)
where 119910 = 119871119894
(119909) 119909 isin [0 119887] then forall119909 isin [119909119894minus1
119909119894
]
119868minusV119891 (119909) = 119865
119894
(119871minus1
119894
(119909) 119868minusV119891 (119871minus1
119894
(119909))) (17)
So 119868minusV119891(119909) is a fractal interpolation function on the interval
[0 119887]
5 Conclusion
There are three acquired results from the above content in thispaper Firstly the fractional order integral of fractal interpola-tion function is continuous on the interval [0 +infin) Secondlythe fractional order integral of fractal interpolation functioncan be integrated on any closed interval [119886 119887] sub [0 +infin)Finally the fractional order integral of fractal interpolationfunction is still a fractal interpolation function on the interval[0 119887](119887 gt 0)
The fractional order integralrsquos differentiability of fractalinterpolation function and its boxing dimension will beresearched in the future
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research was supported by the Scientific Research Inno-vation Foundation for Graduate Students of Jiangsu Province(no CXZZ13 0686) and the Nanjing Normal UniversityTaizhou College Youth Fund Project (no Q201234)
References
[1] B B Mandelbrot ldquoHow long is the coast of Britain Statisticalself-similarity and fractional dimensionrdquo Science vol 156 no3775 pp 636ndash638 1967
[2] B B Mandelbrot and J W van Ness ldquoFractional Brownianmotions fractional noises and applicationsrdquo SIAM Review vol10 pp 422ndash437 1968
[3] B B Mandelbrot The Fractal Geometry of Nature MacmillanPress New York NY USA
[4] B B Mandelbrot D E Passoja and A J Paullay ldquoFractalcharacter of fracture surfaces of metalsrdquo Nature vol 308 no5961 pp 721ndash722 1984
Abstract and Applied Analysis 5
[5] B B Mandelbrot Fractal Objects form Opportunity andDimension World Publishing Corporation Beijing China1999
[6] M F Barnsley ldquoLecture notes on iterated function systemsrdquo inChaos and Fractals vol 39 of Proceedings Symposia in AppliedMathematics pp 127ndash144 American Mathematical SocietyProvidence RI USA 1989
[7] M F Barnsley ldquoFractal functions and interpolationrdquo Construc-tive Approximation vol 2 no 4 pp 303ndash329 1986
[8] P R Massopust ldquoFractal surfacesrdquo Journal of MathematicalAnalysis and Applications vol 151 no 1 pp 275ndash290 1990
[9] P R Massopust Fractal Functions Fractal Surfaces andWavelets Academic Press Orlando Fla USA 1995
[10] Z G Feng and L Wang ldquo120575-variation properties of fractalinterpolation functionsrdquo Journal of Jiangsu University vol 26no 1 pp 49ndash52 2005 (Chinese)
[11] Z Feng ldquoVariation and Minkowski dimension of fractalinterpolation surfacerdquo Journal of Mathematical Analysis andApplications vol 345 no 1 pp 322ndash334 2008
[12] Z G Feng and Y L Huang ldquoVariation and box-countingdimension of fractal interpolation surfaces based on the fractalinterpolation functionrdquo Chinese Journal of Engineering Mathe-matics vol 29 no 3 pp 393ndash398 2012 (Chinese)
[13] S G Li and J T Wu Fractals and Wavelets Science PressBeijing China 2002
[14] Q W Ran and X Y Tan Wavelet Analysis Fractional FourierTransformation and Application National Defence IndustryPress Beijing China 2002
[15] A P Mark Introduction to Fourier Analysis and WaveletsMachine Press Beijing China 2003
[16] I Podlubny Fractional Differential Equations Academic PressSan Diego Calif USA 1999
[17] M F Barnsley Fractals Everywhere Elsevier Singapore 2ndedition 2009
[18] Z Sha andH J Ruan Fractals and Fitting Zhenjiang UniversityPress Hangzhou China 2005
Definition 2 (see [17]) A ldquohyperbolicrdquo iterated functionsystem consists of a complete metric space (119883 119889) togetherwith a finite set of contraction mappings 119908
for 119894 = 1 2 119899The abbreviation ldquoIFSrdquo is used for ldquoiterated function systemrdquoThe notation for the IFS just announced is 119883 119908
119894
119894 =
1 2 119899 and contractivity factor is 119904 = max119904119894
119894 =
1 2 119899
Definition 3 (see [7]) Let (119909119894
119910119894
) isin 1198772
119894 = 0 1 2 119899
be a set of points where 1199090
lt 1199091
lt 1199092
lt sdot sdot sdot lt 119909119899
Aninterpolation function corresponding to this set of data is acontinuous function 119891 [119909
0
119909119899
] rarr 119877 such that
119891 (119909119894
) = 119910119894
119894 = 1 2 119899 (2)
The points (119909119894
119910119894
) are called the interpolation points It iscalled that the function of119891 interpolates the data and that thegraph of 119891 passes through the interpolation points
Lemma 4 (see [17]) Let 119899 be a positive integer greater than1 Let 119877
2
119908119894
119894 = 1 2 119899 denote the IFS defined aboveassociated with the data set
(119909119894
119910119894
) isin 1198772
119894 = 0 1 2 119899 (3)
Let the vertical scaling factor 119889119894
obey 0 le 119889119894
lt 1 for 119894 =
1 2 119899 Then there is a metric 119889 on 1198772 equivalent to the
Euclidean metric such that the IFS is hyperbolic with respectto 119889 In particular there is a unique nonempty compact set119866 sub 119877
2 such that
119866 =
119899
⋃119894=1
119908119894
(119866) (4)
In particular an IFS of the form 1198772
119908119894
119894 = 1 2 119899 isconsidered where the mapping is an affine transformation ofthe special structure
119908119894
(119909
119910) = (
119886119894
0
119888119894
119889119894
)(119909
119910) + (
119890119894
119891119894
) (5)
The transformations are constrained by the data accord-ing to
119908119894
(1199090
1199100
) = (119909119894minus1
119910119894minus1
) 119908119894
(119909119899
119910119899
) = (119909119894
119910119894
) for 119894 = 1 2 119899
(6)
and 119886119894
119890119894
119888119894
119891119894
can be solved from (5) and (6) in terms of thedata and vertical scaling factor 119889
119894
as follows
119886119894
=119909119894
minus 119909119894minus1
119909119899
minus 1199090
119890119894
=119909119899
119909119894minus1
minus 1199090
119909119894
119909119899
minus 1199090
119888119894
=119910119894
minus 119910119894minus1
119909119899
minus 1199090
minus119889119894
(119910119899
minus 1199100
)
119909119899
minus 1199090
119891119894
=119909119899
119910119894minus1
minus 1199090
119910119894
119909119899
minus 1199090
minus119889119894
(119909119899
1199100
minus 1199090
119910119899
)
119909119899
minus 1199090
(7)
Lemma 5 (see [18]) Suppose 119865 is a set of continuous functionswhich satisfy 119891 [119909
0
119909119899
] rarr 119877 and 119891(1199090
) = 1199100
119891(119909119899
) = 119910119899
The metric is defined by the following formula
119889 (119891 119892) = max 1003816100381610038161003816119891 (119909) minus 119892 (119909)1003816100381610038161003816 119909 isin [119909
0
119909119899
] forall119891 119892 isin 119865
(8)
Then (119865 119889) is a complete metric space Let the real numbers119886119894
119888119894
119890119894
119891119894
be defined by (7) Define a mapping 119879 119865 rarr 119865 by
(119879119891) (119909) = 119888119894
119871minus1
119894
(119909) + 119889119894
119891 (119871minus1
119894
(119909)) + 119891119894
119909 isin [119909119894minus1
119909119894
] 119894 = 1 2 119899
(9)
where 119871119894
[1199090
119909119899
] rarr [119909119894minus1
119909119894
] is the invertible transforma-tion
119871119894
(119909) = 119886119894
119909 + 119890119894
119871minus1
119894
(119909) =119909 minus 119890119894
119886119894
119871minus1
119894
(119909119894minus1
) = 1199090
119871minus1
119894
(119909119894
) = 119909119899
(10)
and then 119879119891 is continuous on the interval [119909119894minus1
119909119894
] and 119879 isa contraction mapping on (119865 119889) so 119879 possesses a unique fixedpoint in 119865 That is there exists a function 119891 isin 119865 such that
(119909 minus 119905)Vminus1 1003816100381610038161003816119891 (119905 + Δ119909) minus 119891 (119905)
1003816100381610038161003816 119889119905)
le1
|Γ (V)|(119909
Vminus1119872Δ119909 + 120576int
119909
0
(119909 minus 119905)Vminus1
119889119905)
=1
|Γ (V)|(119909
Vminus1119872Δ119909 +
120576
V119909V) 997888rarr 0
(13)
where 119872 = max119905isin[0Δ119909]
|119891(119905)| so 119868minusV119891(119909) is a continuous
function on the interval [0 +infin)
Corollary 7 Suppose 119891(119909) is a fractal interpolation functionon the interval [0 +infin) then 119868
minusV119891(119909) is continuous on [0 +infin)
too
Proof Since fractal interpolation function of affine trans-formation is a continuous function on [0 +infin) 119891(119909) iscontinuous on [0 +infin) According to Lemma 6 119868minusV119891(119909) is acontinuous function on [0 +infin) too
Corollary 8 Suppose 119891(119909) is a fractal interpolation functionof affine transformation on the interval [119886 119887](0 lt 119886 lt 119887 lt
+infin) then 119868minusV119891(119909) can be integrated on [119886 119887]
Proof From Corollary 7 and since continuous function onfinite closed interval is an integrated function the result ofCorollary 8 is right
4 Judgement Theorem of FractalInterpolation Functionrsquos FractionalIntegral on [0 119887](119887gt0)
Theorem 9 If 119891(119909) is a fractal interpolation function of affinetransformation on the interval [0 119887](119887 gt 0) then 119868
minusV119891(119909) is
a fractal interpolation function of affine transformation on theinterval [0 119887] too
Proof For all 119887 isin (0 +infin) consider the interval [0 119887] for0 lt V lt 1 119909 isin [119909
119894minus1
119909119894
](119909119894
= (119894119899)119887 119894 = 1 2 119899) Letiterated function system (IFS) be
119871119894
(119909) =1
119899119909 +
119894 minus 1
119899119887
119865119894
(119909 119910) = 119888119894
119909 + 119889119894
119910 + 119891119894
(14)
so119871minus1
119894
(119909) = 119899119909 minus (119894 minus 1) 119887
119868minusV119891 (119909) =
1
Γ (V)int119909
0
(119909 minus 119905)Vminus1
119891 (119905) 119889119905
=1
Γ (V)
119896minus1
sum119894=1
int119909119894
119909119894minus1
(119909 minus 119905)Vminus1
119891 (119905) 119889119905
+1
Γ (V)int119909
119909119896minus1
(119909 minus 119905)Vminus1
119891 (119905) 119889119905
=1
Γ (V)
119896minus1
sum119894=1
int119887
0
(119909 minus1
119899119910 minus
119894 minus 1
119899119887)
Vminus1
times 119891 [119871119894
(119910)]1
119899119889119910
(Let 119905 = 119871119894
(119910))
+1
Γ (V)int119871
minus1
119896(119909)
0
(119909 minus1
119899119910 minus
119896 minus 1
119899119887)
Vminus1
times 119891 [119871119896
(119910)]1
119899119889119910
=1
Γ (V) 119899V119896minus1
sum119894=1
int119887
0
[119899119909 minus 119910 minus (119894 minus 1) 119887]Vminus1
times [119888119894
119910 + 119889119894
119891 (119910) + 119891119894
] 119889119910
+1
Γ (V) 119899Vint119871
minus1
119896(119909)
0
[119899119909 minus 119910 minus (119896 minus 1) 119887]Vminus1
times (119888119896
119910 + 119889119896
119891 (119910) + 119891119896
) 119889119910
4 Abstract and Applied Analysis
=1
Γ (V) 119899V119896minus1
sum119894=1
int119887
0
[119871minus1
119894
(119909) minus 119910]Vminus1
times [119888119894
119910 + 119889119894
119891 (119910) + 119891119894
] 119889119910
+119889119896
Γ (V) 119899Vint119871
minus1
119896(119909)
0
[119871minus1
119896
(119909) minus 119910]Vminus1
119891 (119910) 119889119910
+1
Γ (V) 119899Vint119871
minus1
119896(119909)
0
[119871minus1
119896
(119909) minus 119910]Vminus1
times (119888119896
119910 + 119891119896
) 119889119910
=1
Γ (V) 119899V119896minus1
sum119894=1
int119887
0
[119871minus1
119894
(119909) minus 119910]Vminus1
times [119888119894
119910 + 119889119894
119891 (119910) + 119891119894
] 119889119910
+1
Γ (V) 119899Vint119871
minus1
119896(119909)
0
[119871minus1
119896
(119909) minus 119910]Vminus1
times (119888119896
119910 + 119891119896
) 119889119910 +119889119896
119899V119910
=1
Γ (V) 119899V
times
119896minus1
sum119894=1
[119889119894
int119887
0
(119871minus1
119894
(119909) minus 119910)Vminus1
119891 (119910) 119889119910
minus119887119888119894
V(119871minus1
119894
(119909) minus 119887)Vminus
119888119894
V (V + 1)
times [(119871minus1
119894
(119909) minus 119887)V+1
minus (119871minus1
119894
(119909))V+1
]
+ 119887119891119894
]
+119889119896
Γ (V) 119899Vint119871
minus1
119896(119909)
0
(119871minus1
119896
(119909) minus 119910)Vminus1
119891 (119910) 119889119910
+(119871minus1
119896
(119909))V
Γ (V) 119899V[
119888119896
V (V + 1)119871minus1
119896
(119909) +119891119896
V]
(15)
So 119868minusV119891(119909) is a fractal interpolation function of affine trans-
formation on the interval [0 119887] and its iterated functionsystem (IFS)
119871119894
(119909) =1
119899119909 +
119894 minus 1
119899119887 119894 = 1 2 119899
119865119894
(119909 119910)
=1
Γ (V) 119899V
times
119896minus1
sum119894=1
[119889119894
int119887
0
(119871minus1
119894
(119909) minus 119910)Vminus1
119891 (119910) 119889119910
minus119887119888119894
V(119871minus1
119894
(119909) minus 119887)Vminus
119888119894
V (V + 1)
times [(119871minus1
119894
(119909) minus 119887)V+1
minus (119871minus1
119894
(119909))V+1
] + 119887119891119894
]
+119889119896
Γ (V) 119899Vint119871
minus1
119896(119909)
0
(119871minus1
119896
(119909) minus 119910)Vminus1
119891 (119910) 119889119910
+(119871minus1
119896
(119909))V
Γ (V) 119899V[
119888119896
V (V + 1)119871minus1
119896
(119909) +119891119896
V]
(16)
where 119910 = 119871119894
(119909) 119909 isin [0 119887] then forall119909 isin [119909119894minus1
119909119894
]
119868minusV119891 (119909) = 119865
119894
(119871minus1
119894
(119909) 119868minusV119891 (119871minus1
119894
(119909))) (17)
So 119868minusV119891(119909) is a fractal interpolation function on the interval
[0 119887]
5 Conclusion
There are three acquired results from the above content in thispaper Firstly the fractional order integral of fractal interpola-tion function is continuous on the interval [0 +infin) Secondlythe fractional order integral of fractal interpolation functioncan be integrated on any closed interval [119886 119887] sub [0 +infin)Finally the fractional order integral of fractal interpolationfunction is still a fractal interpolation function on the interval[0 119887](119887 gt 0)
The fractional order integralrsquos differentiability of fractalinterpolation function and its boxing dimension will beresearched in the future
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research was supported by the Scientific Research Inno-vation Foundation for Graduate Students of Jiangsu Province(no CXZZ13 0686) and the Nanjing Normal UniversityTaizhou College Youth Fund Project (no Q201234)
References
[1] B B Mandelbrot ldquoHow long is the coast of Britain Statisticalself-similarity and fractional dimensionrdquo Science vol 156 no3775 pp 636ndash638 1967
[2] B B Mandelbrot and J W van Ness ldquoFractional Brownianmotions fractional noises and applicationsrdquo SIAM Review vol10 pp 422ndash437 1968
[3] B B Mandelbrot The Fractal Geometry of Nature MacmillanPress New York NY USA
[4] B B Mandelbrot D E Passoja and A J Paullay ldquoFractalcharacter of fracture surfaces of metalsrdquo Nature vol 308 no5961 pp 721ndash722 1984
Abstract and Applied Analysis 5
[5] B B Mandelbrot Fractal Objects form Opportunity andDimension World Publishing Corporation Beijing China1999
[6] M F Barnsley ldquoLecture notes on iterated function systemsrdquo inChaos and Fractals vol 39 of Proceedings Symposia in AppliedMathematics pp 127ndash144 American Mathematical SocietyProvidence RI USA 1989
[7] M F Barnsley ldquoFractal functions and interpolationrdquo Construc-tive Approximation vol 2 no 4 pp 303ndash329 1986
[8] P R Massopust ldquoFractal surfacesrdquo Journal of MathematicalAnalysis and Applications vol 151 no 1 pp 275ndash290 1990
[9] P R Massopust Fractal Functions Fractal Surfaces andWavelets Academic Press Orlando Fla USA 1995
[10] Z G Feng and L Wang ldquo120575-variation properties of fractalinterpolation functionsrdquo Journal of Jiangsu University vol 26no 1 pp 49ndash52 2005 (Chinese)
[11] Z Feng ldquoVariation and Minkowski dimension of fractalinterpolation surfacerdquo Journal of Mathematical Analysis andApplications vol 345 no 1 pp 322ndash334 2008
[12] Z G Feng and Y L Huang ldquoVariation and box-countingdimension of fractal interpolation surfaces based on the fractalinterpolation functionrdquo Chinese Journal of Engineering Mathe-matics vol 29 no 3 pp 393ndash398 2012 (Chinese)
[13] S G Li and J T Wu Fractals and Wavelets Science PressBeijing China 2002
[14] Q W Ran and X Y Tan Wavelet Analysis Fractional FourierTransformation and Application National Defence IndustryPress Beijing China 2002
[15] A P Mark Introduction to Fourier Analysis and WaveletsMachine Press Beijing China 2003
[16] I Podlubny Fractional Differential Equations Academic PressSan Diego Calif USA 1999
[17] M F Barnsley Fractals Everywhere Elsevier Singapore 2ndedition 2009
[18] Z Sha andH J Ruan Fractals and Fitting Zhenjiang UniversityPress Hangzhou China 2005
(119909 minus 119905)Vminus1 1003816100381610038161003816119891 (119905 + Δ119909) minus 119891 (119905)
1003816100381610038161003816 119889119905)
le1
|Γ (V)|(119909
Vminus1119872Δ119909 + 120576int
119909
0
(119909 minus 119905)Vminus1
119889119905)
=1
|Γ (V)|(119909
Vminus1119872Δ119909 +
120576
V119909V) 997888rarr 0
(13)
where 119872 = max119905isin[0Δ119909]
|119891(119905)| so 119868minusV119891(119909) is a continuous
function on the interval [0 +infin)
Corollary 7 Suppose 119891(119909) is a fractal interpolation functionon the interval [0 +infin) then 119868
minusV119891(119909) is continuous on [0 +infin)
too
Proof Since fractal interpolation function of affine trans-formation is a continuous function on [0 +infin) 119891(119909) iscontinuous on [0 +infin) According to Lemma 6 119868minusV119891(119909) is acontinuous function on [0 +infin) too
Corollary 8 Suppose 119891(119909) is a fractal interpolation functionof affine transformation on the interval [119886 119887](0 lt 119886 lt 119887 lt
+infin) then 119868minusV119891(119909) can be integrated on [119886 119887]
Proof From Corollary 7 and since continuous function onfinite closed interval is an integrated function the result ofCorollary 8 is right
4 Judgement Theorem of FractalInterpolation Functionrsquos FractionalIntegral on [0 119887](119887gt0)
Theorem 9 If 119891(119909) is a fractal interpolation function of affinetransformation on the interval [0 119887](119887 gt 0) then 119868
minusV119891(119909) is
a fractal interpolation function of affine transformation on theinterval [0 119887] too
Proof For all 119887 isin (0 +infin) consider the interval [0 119887] for0 lt V lt 1 119909 isin [119909
119894minus1
119909119894
](119909119894
= (119894119899)119887 119894 = 1 2 119899) Letiterated function system (IFS) be
119871119894
(119909) =1
119899119909 +
119894 minus 1
119899119887
119865119894
(119909 119910) = 119888119894
119909 + 119889119894
119910 + 119891119894
(14)
so119871minus1
119894
(119909) = 119899119909 minus (119894 minus 1) 119887
119868minusV119891 (119909) =
1
Γ (V)int119909
0
(119909 minus 119905)Vminus1
119891 (119905) 119889119905
=1
Γ (V)
119896minus1
sum119894=1
int119909119894
119909119894minus1
(119909 minus 119905)Vminus1
119891 (119905) 119889119905
+1
Γ (V)int119909
119909119896minus1
(119909 minus 119905)Vminus1
119891 (119905) 119889119905
=1
Γ (V)
119896minus1
sum119894=1
int119887
0
(119909 minus1
119899119910 minus
119894 minus 1
119899119887)
Vminus1
times 119891 [119871119894
(119910)]1
119899119889119910
(Let 119905 = 119871119894
(119910))
+1
Γ (V)int119871
minus1
119896(119909)
0
(119909 minus1
119899119910 minus
119896 minus 1
119899119887)
Vminus1
times 119891 [119871119896
(119910)]1
119899119889119910
=1
Γ (V) 119899V119896minus1
sum119894=1
int119887
0
[119899119909 minus 119910 minus (119894 minus 1) 119887]Vminus1
times [119888119894
119910 + 119889119894
119891 (119910) + 119891119894
] 119889119910
+1
Γ (V) 119899Vint119871
minus1
119896(119909)
0
[119899119909 minus 119910 minus (119896 minus 1) 119887]Vminus1
times (119888119896
119910 + 119889119896
119891 (119910) + 119891119896
) 119889119910
4 Abstract and Applied Analysis
=1
Γ (V) 119899V119896minus1
sum119894=1
int119887
0
[119871minus1
119894
(119909) minus 119910]Vminus1
times [119888119894
119910 + 119889119894
119891 (119910) + 119891119894
] 119889119910
+119889119896
Γ (V) 119899Vint119871
minus1
119896(119909)
0
[119871minus1
119896
(119909) minus 119910]Vminus1
119891 (119910) 119889119910
+1
Γ (V) 119899Vint119871
minus1
119896(119909)
0
[119871minus1
119896
(119909) minus 119910]Vminus1
times (119888119896
119910 + 119891119896
) 119889119910
=1
Γ (V) 119899V119896minus1
sum119894=1
int119887
0
[119871minus1
119894
(119909) minus 119910]Vminus1
times [119888119894
119910 + 119889119894
119891 (119910) + 119891119894
] 119889119910
+1
Γ (V) 119899Vint119871
minus1
119896(119909)
0
[119871minus1
119896
(119909) minus 119910]Vminus1
times (119888119896
119910 + 119891119896
) 119889119910 +119889119896
119899V119910
=1
Γ (V) 119899V
times
119896minus1
sum119894=1
[119889119894
int119887
0
(119871minus1
119894
(119909) minus 119910)Vminus1
119891 (119910) 119889119910
minus119887119888119894
V(119871minus1
119894
(119909) minus 119887)Vminus
119888119894
V (V + 1)
times [(119871minus1
119894
(119909) minus 119887)V+1
minus (119871minus1
119894
(119909))V+1
]
+ 119887119891119894
]
+119889119896
Γ (V) 119899Vint119871
minus1
119896(119909)
0
(119871minus1
119896
(119909) minus 119910)Vminus1
119891 (119910) 119889119910
+(119871minus1
119896
(119909))V
Γ (V) 119899V[
119888119896
V (V + 1)119871minus1
119896
(119909) +119891119896
V]
(15)
So 119868minusV119891(119909) is a fractal interpolation function of affine trans-
formation on the interval [0 119887] and its iterated functionsystem (IFS)
119871119894
(119909) =1
119899119909 +
119894 minus 1
119899119887 119894 = 1 2 119899
119865119894
(119909 119910)
=1
Γ (V) 119899V
times
119896minus1
sum119894=1
[119889119894
int119887
0
(119871minus1
119894
(119909) minus 119910)Vminus1
119891 (119910) 119889119910
minus119887119888119894
V(119871minus1
119894
(119909) minus 119887)Vminus
119888119894
V (V + 1)
times [(119871minus1
119894
(119909) minus 119887)V+1
minus (119871minus1
119894
(119909))V+1
] + 119887119891119894
]
+119889119896
Γ (V) 119899Vint119871
minus1
119896(119909)
0
(119871minus1
119896
(119909) minus 119910)Vminus1
119891 (119910) 119889119910
+(119871minus1
119896
(119909))V
Γ (V) 119899V[
119888119896
V (V + 1)119871minus1
119896
(119909) +119891119896
V]
(16)
where 119910 = 119871119894
(119909) 119909 isin [0 119887] then forall119909 isin [119909119894minus1
119909119894
]
119868minusV119891 (119909) = 119865
119894
(119871minus1
119894
(119909) 119868minusV119891 (119871minus1
119894
(119909))) (17)
So 119868minusV119891(119909) is a fractal interpolation function on the interval
[0 119887]
5 Conclusion
There are three acquired results from the above content in thispaper Firstly the fractional order integral of fractal interpola-tion function is continuous on the interval [0 +infin) Secondlythe fractional order integral of fractal interpolation functioncan be integrated on any closed interval [119886 119887] sub [0 +infin)Finally the fractional order integral of fractal interpolationfunction is still a fractal interpolation function on the interval[0 119887](119887 gt 0)
The fractional order integralrsquos differentiability of fractalinterpolation function and its boxing dimension will beresearched in the future
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research was supported by the Scientific Research Inno-vation Foundation for Graduate Students of Jiangsu Province(no CXZZ13 0686) and the Nanjing Normal UniversityTaizhou College Youth Fund Project (no Q201234)
References
[1] B B Mandelbrot ldquoHow long is the coast of Britain Statisticalself-similarity and fractional dimensionrdquo Science vol 156 no3775 pp 636ndash638 1967
[2] B B Mandelbrot and J W van Ness ldquoFractional Brownianmotions fractional noises and applicationsrdquo SIAM Review vol10 pp 422ndash437 1968
[3] B B Mandelbrot The Fractal Geometry of Nature MacmillanPress New York NY USA
[4] B B Mandelbrot D E Passoja and A J Paullay ldquoFractalcharacter of fracture surfaces of metalsrdquo Nature vol 308 no5961 pp 721ndash722 1984
Abstract and Applied Analysis 5
[5] B B Mandelbrot Fractal Objects form Opportunity andDimension World Publishing Corporation Beijing China1999
[6] M F Barnsley ldquoLecture notes on iterated function systemsrdquo inChaos and Fractals vol 39 of Proceedings Symposia in AppliedMathematics pp 127ndash144 American Mathematical SocietyProvidence RI USA 1989
[7] M F Barnsley ldquoFractal functions and interpolationrdquo Construc-tive Approximation vol 2 no 4 pp 303ndash329 1986
[8] P R Massopust ldquoFractal surfacesrdquo Journal of MathematicalAnalysis and Applications vol 151 no 1 pp 275ndash290 1990
[9] P R Massopust Fractal Functions Fractal Surfaces andWavelets Academic Press Orlando Fla USA 1995
[10] Z G Feng and L Wang ldquo120575-variation properties of fractalinterpolation functionsrdquo Journal of Jiangsu University vol 26no 1 pp 49ndash52 2005 (Chinese)
[11] Z Feng ldquoVariation and Minkowski dimension of fractalinterpolation surfacerdquo Journal of Mathematical Analysis andApplications vol 345 no 1 pp 322ndash334 2008
[12] Z G Feng and Y L Huang ldquoVariation and box-countingdimension of fractal interpolation surfaces based on the fractalinterpolation functionrdquo Chinese Journal of Engineering Mathe-matics vol 29 no 3 pp 393ndash398 2012 (Chinese)
[13] S G Li and J T Wu Fractals and Wavelets Science PressBeijing China 2002
[14] Q W Ran and X Y Tan Wavelet Analysis Fractional FourierTransformation and Application National Defence IndustryPress Beijing China 2002
[15] A P Mark Introduction to Fourier Analysis and WaveletsMachine Press Beijing China 2003
[16] I Podlubny Fractional Differential Equations Academic PressSan Diego Calif USA 1999
[17] M F Barnsley Fractals Everywhere Elsevier Singapore 2ndedition 2009
[18] Z Sha andH J Ruan Fractals and Fitting Zhenjiang UniversityPress Hangzhou China 2005
So 119868minusV119891(119909) is a fractal interpolation function of affine trans-
formation on the interval [0 119887] and its iterated functionsystem (IFS)
119871119894
(119909) =1
119899119909 +
119894 minus 1
119899119887 119894 = 1 2 119899
119865119894
(119909 119910)
=1
Γ (V) 119899V
times
119896minus1
sum119894=1
[119889119894
int119887
0
(119871minus1
119894
(119909) minus 119910)Vminus1
119891 (119910) 119889119910
minus119887119888119894
V(119871minus1
119894
(119909) minus 119887)Vminus
119888119894
V (V + 1)
times [(119871minus1
119894
(119909) minus 119887)V+1
minus (119871minus1
119894
(119909))V+1
] + 119887119891119894
]
+119889119896
Γ (V) 119899Vint119871
minus1
119896(119909)
0
(119871minus1
119896
(119909) minus 119910)Vminus1
119891 (119910) 119889119910
+(119871minus1
119896
(119909))V
Γ (V) 119899V[
119888119896
V (V + 1)119871minus1
119896
(119909) +119891119896
V]
(16)
where 119910 = 119871119894
(119909) 119909 isin [0 119887] then forall119909 isin [119909119894minus1
119909119894
]
119868minusV119891 (119909) = 119865
119894
(119871minus1
119894
(119909) 119868minusV119891 (119871minus1
119894
(119909))) (17)
So 119868minusV119891(119909) is a fractal interpolation function on the interval
[0 119887]
5 Conclusion
There are three acquired results from the above content in thispaper Firstly the fractional order integral of fractal interpola-tion function is continuous on the interval [0 +infin) Secondlythe fractional order integral of fractal interpolation functioncan be integrated on any closed interval [119886 119887] sub [0 +infin)Finally the fractional order integral of fractal interpolationfunction is still a fractal interpolation function on the interval[0 119887](119887 gt 0)
The fractional order integralrsquos differentiability of fractalinterpolation function and its boxing dimension will beresearched in the future
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research was supported by the Scientific Research Inno-vation Foundation for Graduate Students of Jiangsu Province(no CXZZ13 0686) and the Nanjing Normal UniversityTaizhou College Youth Fund Project (no Q201234)
References
[1] B B Mandelbrot ldquoHow long is the coast of Britain Statisticalself-similarity and fractional dimensionrdquo Science vol 156 no3775 pp 636ndash638 1967
[2] B B Mandelbrot and J W van Ness ldquoFractional Brownianmotions fractional noises and applicationsrdquo SIAM Review vol10 pp 422ndash437 1968
[3] B B Mandelbrot The Fractal Geometry of Nature MacmillanPress New York NY USA
[4] B B Mandelbrot D E Passoja and A J Paullay ldquoFractalcharacter of fracture surfaces of metalsrdquo Nature vol 308 no5961 pp 721ndash722 1984
Abstract and Applied Analysis 5
[5] B B Mandelbrot Fractal Objects form Opportunity andDimension World Publishing Corporation Beijing China1999
[6] M F Barnsley ldquoLecture notes on iterated function systemsrdquo inChaos and Fractals vol 39 of Proceedings Symposia in AppliedMathematics pp 127ndash144 American Mathematical SocietyProvidence RI USA 1989
[7] M F Barnsley ldquoFractal functions and interpolationrdquo Construc-tive Approximation vol 2 no 4 pp 303ndash329 1986
[8] P R Massopust ldquoFractal surfacesrdquo Journal of MathematicalAnalysis and Applications vol 151 no 1 pp 275ndash290 1990
[9] P R Massopust Fractal Functions Fractal Surfaces andWavelets Academic Press Orlando Fla USA 1995
[10] Z G Feng and L Wang ldquo120575-variation properties of fractalinterpolation functionsrdquo Journal of Jiangsu University vol 26no 1 pp 49ndash52 2005 (Chinese)
[11] Z Feng ldquoVariation and Minkowski dimension of fractalinterpolation surfacerdquo Journal of Mathematical Analysis andApplications vol 345 no 1 pp 322ndash334 2008
[12] Z G Feng and Y L Huang ldquoVariation and box-countingdimension of fractal interpolation surfaces based on the fractalinterpolation functionrdquo Chinese Journal of Engineering Mathe-matics vol 29 no 3 pp 393ndash398 2012 (Chinese)
[13] S G Li and J T Wu Fractals and Wavelets Science PressBeijing China 2002
[14] Q W Ran and X Y Tan Wavelet Analysis Fractional FourierTransformation and Application National Defence IndustryPress Beijing China 2002
[15] A P Mark Introduction to Fourier Analysis and WaveletsMachine Press Beijing China 2003
[16] I Podlubny Fractional Differential Equations Academic PressSan Diego Calif USA 1999
[17] M F Barnsley Fractals Everywhere Elsevier Singapore 2ndedition 2009
[18] Z Sha andH J Ruan Fractals and Fitting Zhenjiang UniversityPress Hangzhou China 2005
[5] B B Mandelbrot Fractal Objects form Opportunity andDimension World Publishing Corporation Beijing China1999
[6] M F Barnsley ldquoLecture notes on iterated function systemsrdquo inChaos and Fractals vol 39 of Proceedings Symposia in AppliedMathematics pp 127ndash144 American Mathematical SocietyProvidence RI USA 1989
[7] M F Barnsley ldquoFractal functions and interpolationrdquo Construc-tive Approximation vol 2 no 4 pp 303ndash329 1986
[8] P R Massopust ldquoFractal surfacesrdquo Journal of MathematicalAnalysis and Applications vol 151 no 1 pp 275ndash290 1990
[9] P R Massopust Fractal Functions Fractal Surfaces andWavelets Academic Press Orlando Fla USA 1995
[10] Z G Feng and L Wang ldquo120575-variation properties of fractalinterpolation functionsrdquo Journal of Jiangsu University vol 26no 1 pp 49ndash52 2005 (Chinese)
[11] Z Feng ldquoVariation and Minkowski dimension of fractalinterpolation surfacerdquo Journal of Mathematical Analysis andApplications vol 345 no 1 pp 322ndash334 2008
[12] Z G Feng and Y L Huang ldquoVariation and box-countingdimension of fractal interpolation surfaces based on the fractalinterpolation functionrdquo Chinese Journal of Engineering Mathe-matics vol 29 no 3 pp 393ndash398 2012 (Chinese)
[13] S G Li and J T Wu Fractals and Wavelets Science PressBeijing China 2002
[14] Q W Ran and X Y Tan Wavelet Analysis Fractional FourierTransformation and Application National Defence IndustryPress Beijing China 2002
[15] A P Mark Introduction to Fourier Analysis and WaveletsMachine Press Beijing China 2003
[16] I Podlubny Fractional Differential Equations Academic PressSan Diego Calif USA 1999
[17] M F Barnsley Fractals Everywhere Elsevier Singapore 2ndedition 2009
[18] Z Sha andH J Ruan Fractals and Fitting Zhenjiang UniversityPress Hangzhou China 2005
