research article new approach to fractal approximation of...
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Research ArticleNew Approach to Fractal Approximation of Vector-Functions
Konstantin Igudesman1 Marsel Davletbaev2 and Gleb Shabernev3
1Geometry Department Lobachevskii Institute of Mathematics and Mechanics Kazan (Volga Region) Federal UniversityKazan 420008 Russia2Kazan (Volga Region) Federal University IT-Lyceum of Kazan University Kazan 420008 Russia3Department of Autonomous Robotic Systems High School of Information Technologies and Information SystemsKazan (Volga Region) Federal University Kazan 420008 Russia
Correspondence should be addressed to Konstantin Igudesman kigudesmyandexru
Received 19 August 2014 Revised 24 January 2015 Accepted 8 February 2015
Academic Editor Poom Kumam
Copyright copy 2015 Konstantin Igudesman et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
This paper introduces new approach to approximation of continuous vector-functions and vector sequences by fractal interpolationvector-functions which are multidimensional generalization of fractal interpolation functions Best values of fractal interpolationvector-functions parameters are found We give schemes of approximation of some sets of data and consider examples ofapproximation of smooth curves with different conditions
1 Introduction
It is well known that interpolation and approximation arean important tool for interpretation of some complicateddata But there are multitudes of interpolationmethods usingseveral families of functions polynomial exponential ratio-nal trigonometric and splines to name a few Still it shouldbe noted that all these conventional nonrecursive methodsproduce interpolants that are differentiable a number of timesexcept possibly at a finite set of points But inmany situationswe deal with irregular forms which can not be approximatewith desired precision Fractal approximation became asuitable tool for that purpose This tool was developed andstudied in [1ndash3]
We know that such curves as coastlines price graphsencephalograms and many others are fractals since theirHausdorff-Besicovitch dimension is greater than unity Toapproximate them we use fractal interpolation curves [1]and their generalizations [4] instead of canonical smoothfunctions (polynomials and splines)
This paper is multidimensional generalization of [5] InSection 2 we consider fractal interpolation vector-functionswhich depend on several matrices of parameters Example of
such functions is given In Section 3 we set the optimizationproblem for approximation of vector-function from 119871
2by
fractal approximation vector-functions We find best valuesofmatrix parameters bymeans ofmatrix differential calculusSection 4 illustrates some examples
2 Fractal Interpolation Vector-Functions
Let [119886 119887] sub R be a nonempty interval let 1 lt 119873 isin N and(119905119899 x119899) isin [119886 119887] times R119872 | 119886 = 119905
0lt 1199051lt sdot sdot sdot lt 119905
119873minus1lt 119905119873= 119887
be the interpolation points For all 119899 = 1119873 consider affinetransformation
119860119899 R119872+1
997888rarr R119872+1
119860119899(
119905
x) = (
119886119899
0c119899
D119899
)(
119905
x) + (
119890119899
f119899
)
(1)
Henceforth small bold letters denote columns (rows) oflength119872 and big bold letters denote matrices of119872times119872
Require that for all 119899 the following conditions hold true
119860119899(1199050 x0) = (119905
119899minus1 x119899minus1
) 119860119899(119905119873 x119873) = (119905
119899 x119899) (2)
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2015 Article ID 278313 7 pageshttpdxdoiorg1011552015278313
2 Abstract and Applied Analysis
Then
1198861198991199050+ 119890119899= 119905119899minus1
119886119899119905119873+ 119890119899= 119905119899
c1198991199050+D119899x0+ f119899= x119899minus1
c119899119905119873+D119899x119873+ f119899= x119899
(3)
Solving the system we have
119886119899=119905119899minus 119905119899minus1
119887 minus 119886
119890119899=119887119905119899minus1
minus 119886119905119899
119887 minus 119886
c119899=x119899minus x119899minus1
minusD119899(x119873minus x0)
119887 minus 119886
f119899=119887x119899minus1
minus 119886x119899minusD119899(119887x0minus 119886x119873)
119887 minus 119886
(4)
where matrices D119899119873
119899=1are considered as parameters
Remark 1 Notice that sum119873119899=1
119886119899= 1
Also notice that for all 119899 operator 119860119899takes straight seg-
ment between (1199050 x0) and (119905
119873 x119873) to straight segment which
connects points of interpolation (119905119899minus1
x119899minus1
) and (119905119899 x119899)
LetK be a space of nonempty compact subsets of R119872+1with Hausdorff metric Define the Hutchinson operator [6]
Φ K 997888rarr K Φ (119864) =
119873
⋃
119899=1
119860119899(119864) (5)
By the condition (2) Hutchinson operator Φ takes a graph ofany continuous vector-function on segment [119886 119887] to a graphof a continuous vector-function on the same segment ThusΦ can be treated as operator on the space of continuousvector-functions (119862[119886 119887])119872
For all 119899 = 1119873 denote
119901119899 [119886 119887] 997888rarr [119905
119899minus1 119905119899] 119901
119899(119905) = 119886
119899119905 + 119890119899
q119899 [119886 119887] 997888rarr R
119872 q119899(119905) = c
119899119905 + f119899
(6)
In (1) substitute x to vector-function g(119905)We have thatΦ actson (119862[119886 119887])119872 according to
(Φg) (119905)
=
119873
sum
119899=1
((q119899∘ 119901minus1
119899) (119905) +D
119899(g ∘ 119901minus1
119899) (119905)) 120594
[119905119899minus1119905119899](119905)
(7)
Suppose that we consider all matricesD119899as linear opera-
tors onR119872 Furthermore they are contractivemappings thatis constant 119888 isin [0 1) exists such that for all kw isin R119872 and119899 = 1119873 we have
1003816100381610038161003816D119899 (k) minusD119899(w)1003816100381610038161003816 le 119888 |k minus w| (8)
Then from (7) it follows that operator Φ is contraction withcontraction coefficient 119888 on Banach space ((119862[119886 119887])119872 sdot
infin)
where g(119905) minus h(119905)infin
= sup119905 isin [119886 119887] |g(119905) minus h(119905)| Bythe fixed-point theorem there exists unique vector-functiong⋆ isin (119862[119886 119887])119872 such thatΦg⋆ = g⋆ and for all g isin (119862[119886 119887])119872we have
lim119896rarrinfin
10038171003817100381710038171003817Φ119896(g) minus g⋆10038171003817100381710038171003817infin = 0 (9)
Function g⋆ is called fractal interpolation vector-functionIt is easy to notice that if g isin (119862[119886 119887])
119872 g(1199050) = x
0 and
g(119905119873) = x119873 thenΦ(g) passes through points of interpolation
In this case functionsΦ119896(g) are called prefractal interpolationvector-functions of order 119896
Example 2 Figure 1 shows fractal interpolation vector-function of plane Here 119905
0= minus1 119905
1= 0 and 119905
2= 1 and
1199090= (1 minus1) 119909
1= (0 0) and 119909
2= (1 1) Values of matrices
D1andD
2are
(
minus1
4
3
4
minus1
4
1
2
) (
minus1
4minus3
4
1
4
1
2
) (10)
3 Approximation
Henceforth we assume that for all 119899 = 1119873 linear operatorD119899is contractive mapping with contraction coefficient 119888 isin
[0 1) We approximate vector-function g isin (119862[119886 119887])119872
by fractal interpolation vector-function g⋆ constructed onpoints of interpolation (119905
119899 x119899)119873
119899=0 Thus we need to fit
matrix parameters D119899to minimize the distance between g
and g⋆We use methods that have been developed for fractal
image compression [7] Denote Banach space of squareintegrated vector-functions on segment as (119871119872
2[119886 119887] sdot
2)
where norm sdot 2defines
1003817100381710038171003817g10038171003817100381710038172= radicint
119887
119886
1003816100381610038161003816g (119905)1003816100381610038161003816
2 d119905 (11)
Then from (7) and (8) and Remark 1 it follows that for allg h isin 119871119872
2[119886 119887]
1003817100381710038171003817Φg minus Φh1003817100381710038171003817
2
2
= int
119887
119886
1003816100381610038161003816Φg minus Φh1003816100381610038161003816
2 d119905
=
119873
sum
119899=1
int
119905119899
119905119899minus1
10038161003816100381610038161003816D119899∘ (g minus h) ∘ 119906minus1
119899(119905)10038161003816100381610038161003816
2
d119905
=
119873
sum
119899=1
119886119899int
119887
119886
1003816100381610038161003816D119899 ∘ (g minus h) (119905)10038161003816100381610038162 d119905
le
119873
sum
119899=1
1198861198991198882int
119887
119886
1003816100381610038161003816(g minus h) (119905)10038161003816100381610038162 d119905 = 1198882 1003817100381710038171003817g minus h1003817100381710038171003817
2
2
(12)
Abstract and Applied Analysis 3
10
05
minus05
minus10
02 04 06 08 10
Figure 1 Fractal interpolation vector-function g⋆
Thus Φ 119871119872
2[119886 119887] rarr 119871
119872
2[119886 119887] is a contractive operator and
g⋆ is its fixed pointInstead of minimizing g minus g⋆
2we minimize g minus Φg
2
that makes the problem of optimization much easier Thecollage theorem provides validity of such approach [8]
Theorem 3 Let (119883 119889) be complete metric space and 119879 119883 rarr
119883 is contractive mapping with contraction coefficient 119888 isin [0 1)and fixed point 119909⋆ Then
119889 (119909 119909⋆) le
119889 (119909 119879 (119909))
1 minus 119888(13)
for all 119909 isin 119883
Considering (4) and (6) rewrite (7)
(Φg) (119905) =119873
sum
119899=1
(u119899(119905) +D
119899(g ∘ 119908
119899(119905) minus k
119899(119905))) 120594
[119905119899minus1119905119899](119905)
(14)
where
u119899(119905) =
(x119899minus x119899minus1
) 119905 + (119905119899x119899minus1
minus 119905119899minus1
x119899)
119905119899minus 119905119899minus1
k119899(119905) =
(x119873minus x0) 119905 + (119905
119899x0minus 119905119899minus1
x119873)
119905119899minus 119905119899minus1
119908119899(119905) =
(119887 minus 119886) 119905 + (119905119899119886 minus 119905119899minus1
119887)
119905119899minus 119905119899minus1
(15)
Thus we minimize the functional
1003817100381710038171003817g minus Φg1003817100381710038171003817
2
2
=
119873
sum
119899=1
int
119905119899
119905119899minus1
1003816100381610038161003816g (119905) minus u119899(119905) minusD
119899(g ∘ 119908
119899(119905) minus k
119899(119905))
1003816100381610038161003816
2 d119905(16)
Lemma 4 Let f h isin 119871119872
2[119886 119887] be square integrated vector-
functions Suppose that matrix int119887
119886hh119879dt is nondegenerated
Matrix integration is implied to be componentwise Then thefunctional
Ψ R119872times119872
997888rarr R Ψ (X) = int119887
119886
|f minus Xh|2 dt (17)
reaches its minimum in X = int119887
119886fh119879dt (intba hhTdt)minus1
Proof To prove it we use matrix differential calculus [9]Consider
dΨ (XU) = d(int119887
119886
(f minus Xh)119879 (f minus Xh) d119905)U
= d(int119887
119886
(f119879f minus h119879X119879f minus f119879Xh + h119879X119879Xh) d119905)U
= int
119887
119886
(minush119879U119879f minus f119879Uh + h119879U119879Xh + h119879X119879Uh) d119905
= 2int
119887
119886
(minush119879U119879f + h119879U119879Xh) d119905(18)
Necessary condition of existence of functional Ψ extremumis dΨ(XU) = 0 for all U isin R119872times119872 Since there is 119880-linearityof functional dΨ(XU) it is sufficient to prove dΨ(XU) = 0
only formatricesU that consist of1198722minus1 zeros and one unityTherefore we have1198722 expressions for finding coefficients ofmatrix X In matrix form these expressions are as follows
int
119887
119886
fh119879d119905 = int119887
119886
Xhh119879d119905 (19)
from which
X = int
119887
119886
fh119879d119905 (int119887
119886
hh119879d119905)minus1
(20)
Hence
d2Ψ (XU) = 2int119887
119886
h119879U119879Uh d119905 = 2int119887
119886
|Uh|2 d119905 ge 0 (21)
and then functional Ψ is convex one Thus the value X isabsolute minimum of Ψ
From Lemma 4 it follows that functional (16) reachesminimum when
D119899= int
119905119899
119905119899minus1
(g (119905) minus u119899(119905)) (g ∘ 119908
119899(119905) minus k
119899(119905))119879 d119905
sdot (int
119905119899
119905119899minus1
(g ∘ 119908119899(119905) minus k
119899(119905)) (g ∘ 119908
119899(119905) minus k
119899(119905))119879 d119905)minus1
(22)
4 Abstract and Applied Analysis
1
09
08
07
06
05
04
03
02
01
7644e minus 31
minus1
minus08
minus06
minus04
minus02
02
04
06
08 1
2165eminus15
Figure 2 Vector-function g(119905) = (1199052 1199053) and fractal interpolation
vector-function g⋆ completely identical
Example 5 Let us approximate vector-function g(119905) =
(1199052 1199053) on segment [minus1 1] by the fractal interpolation vector-
function constructed on values of g(119905) in points 1199050= minus1 119905
1=
0 and 1199052= 1 and 119909
0= (1 minus1) 119909
1= (0 0) and 119909
2= (1 1)
(see Figure 2) Then
1198861= 1198862=1
2
1198901= minus
1
2 119890
2=1
2
u1= (minus119905 119905) u
2= (119905 119905)
k1= (1 1 + 2119905) k
2= (1 minus1 + 2119905)
1199081= 1 + 2119905 119908
2= minus1 + 2119905
2
(23)
CalculateD1D2according to formula (22) as follows
D1= (
1
40
minus3
8
1
8
) D2= (
1
40
3
8
1
8
) (24)
Apply affine transformations from (1) to vector 119905 1199052 1199053
1198601(
119905
1199052
1199053
) =(
(
1
20 0
minus1
2
1
40
3
8minus3
8
1
8
)
)
(
119905
1199052
1199053
)+(
(
minus1
2
1
4
minus1
8
)
)
=((
(
119905
2minus1
2
1199052
4minus119905
2+1
4
1199053
8minus31199052
8+3119905
8minus1
8
))
)
=((
(
119905minus 1
2
(119905 minus 1
2)
2
(119905 minus 1
2)
3
))
)
1198602(
119905
1199052
1199053
) =(
(
1
20 0
1
2
1
40
3
8
3
8
1
8
)
)
(
119905
1199052
1199053
)+(
(
1
2
1
4
1
8
)
)
=((
(
119905
2+1
2
1199052
4+119905
2+1
4
1199053
8+31199052
8+3119905
8+1
8
))
)
=((
(
119905+ 1
2
(119905 + 1
2)
2
(119905 + 1
2)
3
))
)
(25)
Thus Φ(g) = g and g = g⋆
4 Discretization and Results
In this section we approximate discrete data 119885 =
(119911119898w119898)119870
119896=0 119886 = 119911
0lt 1199111lt sdot sdot sdot lt 119911
119870= 119887 by fractal
interpolation vector-function g⋆ constructed on points ofinterpolation 119883 = (119905
119894 x119894)119873
119894=0 119886 = 119905
0lt 1199051lt sdot sdot sdot lt 119905
119873= 119887
119873 ≪ 119870 Assume that119883 sub 119885 We fit matrix parametersD119899to
minimize functional119870
sum
119896=0
1003816100381610038161003816w119896 minus g⋆ (119911119896)1003816100381610038161003816
2
(26)
It is necessary to use results of previous section Approxi-mate 119885 by constant piecewise vector-function g [119886 119887] rarr
R119872 More precisely g(119911) = w119896 where (119911
119896w119896) isin 119885 119911
119896
is the nearest approximation neighbor of 119911 By substitutingintegrals in (22) to discretization points sums we obtain
D119899
= ( sum
119911119896isin[119905119899minus1119905119899]
(g (119911119896) minus u119899(119911119896)) (g ∘ 119908
119899(119911119896) minus k119899(119911119896))119879
)
sdot ( sum
119911119896isin[119905119899minus1119905119899]
(g ∘ 119908119899(119911119896) minus k119899(119911119896))
Abstract and Applied Analysis 5
64
5686
4972
4258
3544
2829
2115
1401
6871
minus02697
minus7411minus1 06 22 38 54 7 86 102 118 134 15
(a)
minus1 06 22 38 54 7 86 102 117 134 15
64
5736
5072
4408
3745
3081
2417
1753
1089
4253
minus2385
(b)
Figure 3 Approximation of vector-function g(119905) = (119905(119905 minus 2) (119905 minus 1)2(119905 + 1)
2) by fractal interpolation function g⋆ with three (a) and four (b)
points of interpolation correspondingly
sdot (g ∘ 119908119899(119911119896) minus k119899(119911119896))119879
)
minus1
119899 = 1119873
(27)
It is sufficient to apply (1) for constructing fractal interpola-tion vector-function after we findD
119899
Consider several examples of approximation of discretedata
Example 6 Let us approximate vector-function g(119905) = (119905(119905 minus
2) (119905 minus 1)2(119905 + 1)
2) where 119905 isin [minus3 3] Figure 3 shows the
results Here we have two pictures the first one illustratesinitial vector-function and its approximation with 3 pointsand the second one with 4 points where two functions arenearly identical
In this case affine transformations (1) have the followingform
1198601(
119905
1199091
1199092
) = (
05 0 0
minus08842 00943 minus01045
minus01038 minus00530 02287
)(
119905
1199091
1199092
)
+(
0
07320
53989
)
1198602(
119905
1199091
1199092
) = (
05 0 0
34602 07065 minus15549
minus04554 01504 minus02847
)(
119905
1199091
1199092
)
+(
075
108875
81844
)
(28)
Remark 7 Vectors c119899in matrices of affine transformations
(1) equal 0 (like in previous example) It means that fractalinterpolation vector-function can be treated as attractor ofclassical affine IFS in R119872
Example 8 Next example is devoted to a circle g(119905) =
(cos 119905 sin 119905) 119905 isin [0 2120587] Figure 4 shows the results Here wealso have two pictures the first one illustrates initial vector-function and its approximation with 3 points and the secondone with 5 points
In this case affine transformations (1) have the followingform
1198601(
119905
1199091
1199092
) = (
05 0 0
minus03180 00006 02128
minus00013 minus05686 minus00038
)(
119905
1199091
1199092
)
+(
0
09993
05686
)
1198602(
119905
1199091
1199092
) = (
05 0 0
03181 minus00053 minus02128
00040 05686 00020
)(
119905
1199091
1199092
)
+(
3151
minus09945
minus05770
)
(29)
Example 9 Spiral of Archimedes g(119905) = (119905 cos 119905 119905 sin 119905) 119905 isin[0 5120587] where the scheme is equal to the examples above buthere we use far more points of interpolation as illustrated inFigure 5
6 Abstract and Applied Analysis
1
08
06
04
02
111e minus 05
minus02
minus04
minus06
minus08
minus1
minus1133
minus09065
minus06798
minus04532
minus02266
4005eminus05
02267
04533
06799
09055
1133
(a)
1
08
06
04
02
111e minus 06
minus02
minus04
minus06
minus08
minus1
minus1
minus08
minus06
minus04
minus22
minus1099eminus06
02
04
06
08 1
(b)
Figure 4 Approximation of vector-function g(119905) = (cos 119905 sin 119905) by fractal interpolation function g⋆ with three (a) and five (b) points ofinterpolation correspondingly
143
1116
8014
4872
173
minus1412
minus4554
minus7696
minus1084
minus1398
minus1712
minus1251
minus9668
minus6831
minus3995
minus1158
1679
4516
7353
1019
1303
1586
(a)
1261
9775
6944
4113
1283
minus1548
minus4379
minus721
minus1004
minus1287
minus157
minus1152
minus8856
minus6191
minus3526
minus08612
1804
4469
7134
9798
1246
15
(b)
Figure 5 Approximation of vector-function g(119905) = (119905 cos 119905 119905 sin 119905) 119905 isin [0 5120587] by fractal interpolation function g⋆ with twelve (a) andseventeen (b) points of interpolation correspondingly
Example 10 Figure 6 shows approximation of vector-function g(119905) = (cos(15119905) sin(119905)) 119905 isin [0 12120587] byfractal interpolation vector-function with sixteen pointsof interpolation
Example 11 The example illustrates approximation of graphof Weierstrass function 120596(119909) = sum
infin
119899=0(12)119899 cos(21205874119899119909)
(Figure 7) by fractal interpolation vector-function
This example is taken from [10] where fractal approxima-tion is used for approximate calculation of box dimension offractal curves
5 Conclusion
In this paper we have introduced new effective method ofapproximation of continuous vector-functions and vector
Abstract and Applied Analysis 7
1021
08172
06132
04092
02051
0001096
minus02029
minus0407
minus0611
minus0815
minus1019
minus1025
minus08177
minus06109
minus0404
minus01972
0009691
02165
04234
06303
08371
1044
Figure 6 Approximation of vector-function g(119905) = (cos(15119905)sin(119905)) by fractal interpolation function g⋆
2
15
1
05
0
minus05
minus1
minus15
minus2
minus1 minus00 minus0e minus04 minus32 0 02 04 06 30 1
Figure 7 Weierstrass function (blue one) and approximatingvector-function (red one)
sequences by fractal interpolation vector-functions whichare affine transformations withmatrix parameters Parameterfitting was a crucial part of approximation process We havefound appropriate parameter values of fractal interpolationvector-functions and illustrate it with several examples ofdifferent types of discrete data
We assume that fractal approximation is highly promisingcomputational tool for different types of data and it canbe used in many ways even in interdisciplinary fieldswith a quite high precision that allows us to apply fractalapproximation methods to a wide variety of curves smoothand nonsmooth alike
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The work was performed according to the Russian Gov-ernment Program of Competitive Growth of Kazan Federal
University The authors greatly acknowledge the anonymousreviewer for carefully reading the paper and providing con-structive comments that have led to an improved paper
References
[1] M F Barnsley ldquoFractal functions and interpolationrdquo Construc-tive Approximation vol 2 no 4 pp 303ndash329 1986
[2] M F Barnsley J Elton D Hardin and P Massopust ldquoHiddenvariable fractal interpolation functionsrdquo SIAM Journal onMath-ematical Analysis vol 20 no 5 pp 1218ndash1242 1989
[3] M F Barnsley and A N Harrington ldquoThe calculus of fractalinterpolation functionsrdquo Journal of Approximation Theory vol57 no 1 pp 14ndash34 1989
[4] P Massopust Interpolation and Approximation with Splines andFractals Oxford University Press Oxford UK 2010
[5] K Igudesman and G Shabernev ldquoNovel method of fractalapproximationrdquo Lobachevskii Journal of Mathematics vol 34no 2 pp 125ndash132 2013
[6] J E Hutchinson ldquoFractals and self-similarityrdquo Indiana Univer-sity Mathematics Journal vol 30 no 5 pp 713ndash747 1981
[7] M F Barnsley and L P Hurd Fractal Image Compression A KPeters Wellesley Mass USA 1993
[8] M Barnsley Fractals Everywhere Academic Press BostonMass USA 1988
[9] J R Magnus and H Neudecker ldquoMatrix differential calculuswith applications in statistics and econometricsrdquoApplied Math-ematical Sciences vol 8 no 144 pp 7175ndash7181 2014
[10] K Igudesman R Lavrenov and V Klassen Novel Approach toCalculation of Box Dimension of Fractal Functions Wiley 3rdedition 2007
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical PhysicsAdvances in
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
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Stochastic AnalysisInternational Journal of
2 Abstract and Applied Analysis
Then
1198861198991199050+ 119890119899= 119905119899minus1
119886119899119905119873+ 119890119899= 119905119899
c1198991199050+D119899x0+ f119899= x119899minus1
c119899119905119873+D119899x119873+ f119899= x119899
(3)
Solving the system we have
119886119899=119905119899minus 119905119899minus1
119887 minus 119886
119890119899=119887119905119899minus1
minus 119886119905119899
119887 minus 119886
c119899=x119899minus x119899minus1
minusD119899(x119873minus x0)
119887 minus 119886
f119899=119887x119899minus1
minus 119886x119899minusD119899(119887x0minus 119886x119873)
119887 minus 119886
(4)
where matrices D119899119873
119899=1are considered as parameters
Remark 1 Notice that sum119873119899=1
119886119899= 1
Also notice that for all 119899 operator 119860119899takes straight seg-
ment between (1199050 x0) and (119905
119873 x119873) to straight segment which
connects points of interpolation (119905119899minus1
x119899minus1
) and (119905119899 x119899)
LetK be a space of nonempty compact subsets of R119872+1with Hausdorff metric Define the Hutchinson operator [6]
Φ K 997888rarr K Φ (119864) =
119873
⋃
119899=1
119860119899(119864) (5)
By the condition (2) Hutchinson operator Φ takes a graph ofany continuous vector-function on segment [119886 119887] to a graphof a continuous vector-function on the same segment ThusΦ can be treated as operator on the space of continuousvector-functions (119862[119886 119887])119872
For all 119899 = 1119873 denote
119901119899 [119886 119887] 997888rarr [119905
119899minus1 119905119899] 119901
119899(119905) = 119886
119899119905 + 119890119899
q119899 [119886 119887] 997888rarr R
119872 q119899(119905) = c
119899119905 + f119899
(6)
In (1) substitute x to vector-function g(119905)We have thatΦ actson (119862[119886 119887])119872 according to
(Φg) (119905)
=
119873
sum
119899=1
((q119899∘ 119901minus1
119899) (119905) +D
119899(g ∘ 119901minus1
119899) (119905)) 120594
[119905119899minus1119905119899](119905)
(7)
Suppose that we consider all matricesD119899as linear opera-
tors onR119872 Furthermore they are contractivemappings thatis constant 119888 isin [0 1) exists such that for all kw isin R119872 and119899 = 1119873 we have
1003816100381610038161003816D119899 (k) minusD119899(w)1003816100381610038161003816 le 119888 |k minus w| (8)
Then from (7) it follows that operator Φ is contraction withcontraction coefficient 119888 on Banach space ((119862[119886 119887])119872 sdot
infin)
where g(119905) minus h(119905)infin
= sup119905 isin [119886 119887] |g(119905) minus h(119905)| Bythe fixed-point theorem there exists unique vector-functiong⋆ isin (119862[119886 119887])119872 such thatΦg⋆ = g⋆ and for all g isin (119862[119886 119887])119872we have
lim119896rarrinfin
10038171003817100381710038171003817Φ119896(g) minus g⋆10038171003817100381710038171003817infin = 0 (9)
Function g⋆ is called fractal interpolation vector-functionIt is easy to notice that if g isin (119862[119886 119887])
119872 g(1199050) = x
0 and
g(119905119873) = x119873 thenΦ(g) passes through points of interpolation
In this case functionsΦ119896(g) are called prefractal interpolationvector-functions of order 119896
Example 2 Figure 1 shows fractal interpolation vector-function of plane Here 119905
0= minus1 119905
1= 0 and 119905
2= 1 and
1199090= (1 minus1) 119909
1= (0 0) and 119909
2= (1 1) Values of matrices
D1andD
2are
(
minus1
4
3
4
minus1
4
1
2
) (
minus1
4minus3
4
1
4
1
2
) (10)
3 Approximation
Henceforth we assume that for all 119899 = 1119873 linear operatorD119899is contractive mapping with contraction coefficient 119888 isin
[0 1) We approximate vector-function g isin (119862[119886 119887])119872
by fractal interpolation vector-function g⋆ constructed onpoints of interpolation (119905
119899 x119899)119873
119899=0 Thus we need to fit
matrix parameters D119899to minimize the distance between g
and g⋆We use methods that have been developed for fractal
image compression [7] Denote Banach space of squareintegrated vector-functions on segment as (119871119872
2[119886 119887] sdot
2)
where norm sdot 2defines
1003817100381710038171003817g10038171003817100381710038172= radicint
119887
119886
1003816100381610038161003816g (119905)1003816100381610038161003816
2 d119905 (11)
Then from (7) and (8) and Remark 1 it follows that for allg h isin 119871119872
2[119886 119887]
1003817100381710038171003817Φg minus Φh1003817100381710038171003817
2
2
= int
119887
119886
1003816100381610038161003816Φg minus Φh1003816100381610038161003816
2 d119905
=
119873
sum
119899=1
int
119905119899
119905119899minus1
10038161003816100381610038161003816D119899∘ (g minus h) ∘ 119906minus1
119899(119905)10038161003816100381610038161003816
2
d119905
=
119873
sum
119899=1
119886119899int
119887
119886
1003816100381610038161003816D119899 ∘ (g minus h) (119905)10038161003816100381610038162 d119905
le
119873
sum
119899=1
1198861198991198882int
119887
119886
1003816100381610038161003816(g minus h) (119905)10038161003816100381610038162 d119905 = 1198882 1003817100381710038171003817g minus h1003817100381710038171003817
2
2
(12)
Abstract and Applied Analysis 3
10
05
minus05
minus10
02 04 06 08 10
Figure 1 Fractal interpolation vector-function g⋆
Thus Φ 119871119872
2[119886 119887] rarr 119871
119872
2[119886 119887] is a contractive operator and
g⋆ is its fixed pointInstead of minimizing g minus g⋆
2we minimize g minus Φg
2
that makes the problem of optimization much easier Thecollage theorem provides validity of such approach [8]
Theorem 3 Let (119883 119889) be complete metric space and 119879 119883 rarr
119883 is contractive mapping with contraction coefficient 119888 isin [0 1)and fixed point 119909⋆ Then
119889 (119909 119909⋆) le
119889 (119909 119879 (119909))
1 minus 119888(13)
for all 119909 isin 119883
Considering (4) and (6) rewrite (7)
(Φg) (119905) =119873
sum
119899=1
(u119899(119905) +D
119899(g ∘ 119908
119899(119905) minus k
119899(119905))) 120594
[119905119899minus1119905119899](119905)
(14)
where
u119899(119905) =
(x119899minus x119899minus1
) 119905 + (119905119899x119899minus1
minus 119905119899minus1
x119899)
119905119899minus 119905119899minus1
k119899(119905) =
(x119873minus x0) 119905 + (119905
119899x0minus 119905119899minus1
x119873)
119905119899minus 119905119899minus1
119908119899(119905) =
(119887 minus 119886) 119905 + (119905119899119886 minus 119905119899minus1
119887)
119905119899minus 119905119899minus1
(15)
Thus we minimize the functional
1003817100381710038171003817g minus Φg1003817100381710038171003817
2
2
=
119873
sum
119899=1
int
119905119899
119905119899minus1
1003816100381610038161003816g (119905) minus u119899(119905) minusD
119899(g ∘ 119908
119899(119905) minus k
119899(119905))
1003816100381610038161003816
2 d119905(16)
Lemma 4 Let f h isin 119871119872
2[119886 119887] be square integrated vector-
functions Suppose that matrix int119887
119886hh119879dt is nondegenerated
Matrix integration is implied to be componentwise Then thefunctional
Ψ R119872times119872
997888rarr R Ψ (X) = int119887
119886
|f minus Xh|2 dt (17)
reaches its minimum in X = int119887
119886fh119879dt (intba hhTdt)minus1
Proof To prove it we use matrix differential calculus [9]Consider
dΨ (XU) = d(int119887
119886
(f minus Xh)119879 (f minus Xh) d119905)U
= d(int119887
119886
(f119879f minus h119879X119879f minus f119879Xh + h119879X119879Xh) d119905)U
= int
119887
119886
(minush119879U119879f minus f119879Uh + h119879U119879Xh + h119879X119879Uh) d119905
= 2int
119887
119886
(minush119879U119879f + h119879U119879Xh) d119905(18)
Necessary condition of existence of functional Ψ extremumis dΨ(XU) = 0 for all U isin R119872times119872 Since there is 119880-linearityof functional dΨ(XU) it is sufficient to prove dΨ(XU) = 0
only formatricesU that consist of1198722minus1 zeros and one unityTherefore we have1198722 expressions for finding coefficients ofmatrix X In matrix form these expressions are as follows
int
119887
119886
fh119879d119905 = int119887
119886
Xhh119879d119905 (19)
from which
X = int
119887
119886
fh119879d119905 (int119887
119886
hh119879d119905)minus1
(20)
Hence
d2Ψ (XU) = 2int119887
119886
h119879U119879Uh d119905 = 2int119887
119886
|Uh|2 d119905 ge 0 (21)
and then functional Ψ is convex one Thus the value X isabsolute minimum of Ψ
From Lemma 4 it follows that functional (16) reachesminimum when
D119899= int
119905119899
119905119899minus1
(g (119905) minus u119899(119905)) (g ∘ 119908
119899(119905) minus k
119899(119905))119879 d119905
sdot (int
119905119899
119905119899minus1
(g ∘ 119908119899(119905) minus k
119899(119905)) (g ∘ 119908
119899(119905) minus k
119899(119905))119879 d119905)minus1
(22)
4 Abstract and Applied Analysis
1
09
08
07
06
05
04
03
02
01
7644e minus 31
minus1
minus08
minus06
minus04
minus02
02
04
06
08 1
2165eminus15
Figure 2 Vector-function g(119905) = (1199052 1199053) and fractal interpolation
vector-function g⋆ completely identical
Example 5 Let us approximate vector-function g(119905) =
(1199052 1199053) on segment [minus1 1] by the fractal interpolation vector-
function constructed on values of g(119905) in points 1199050= minus1 119905
1=
0 and 1199052= 1 and 119909
0= (1 minus1) 119909
1= (0 0) and 119909
2= (1 1)
(see Figure 2) Then
1198861= 1198862=1
2
1198901= minus
1
2 119890
2=1
2
u1= (minus119905 119905) u
2= (119905 119905)
k1= (1 1 + 2119905) k
2= (1 minus1 + 2119905)
1199081= 1 + 2119905 119908
2= minus1 + 2119905
2
(23)
CalculateD1D2according to formula (22) as follows
D1= (
1
40
minus3
8
1
8
) D2= (
1
40
3
8
1
8
) (24)
Apply affine transformations from (1) to vector 119905 1199052 1199053
1198601(
119905
1199052
1199053
) =(
(
1
20 0
minus1
2
1
40
3
8minus3
8
1
8
)
)
(
119905
1199052
1199053
)+(
(
minus1
2
1
4
minus1
8
)
)
=((
(
119905
2minus1
2
1199052
4minus119905
2+1
4
1199053
8minus31199052
8+3119905
8minus1
8
))
)
=((
(
119905minus 1
2
(119905 minus 1
2)
2
(119905 minus 1
2)
3
))
)
1198602(
119905
1199052
1199053
) =(
(
1
20 0
1
2
1
40
3
8
3
8
1
8
)
)
(
119905
1199052
1199053
)+(
(
1
2
1
4
1
8
)
)
=((
(
119905
2+1
2
1199052
4+119905
2+1
4
1199053
8+31199052
8+3119905
8+1
8
))
)
=((
(
119905+ 1
2
(119905 + 1
2)
2
(119905 + 1
2)
3
))
)
(25)
Thus Φ(g) = g and g = g⋆
4 Discretization and Results
In this section we approximate discrete data 119885 =
(119911119898w119898)119870
119896=0 119886 = 119911
0lt 1199111lt sdot sdot sdot lt 119911
119870= 119887 by fractal
interpolation vector-function g⋆ constructed on points ofinterpolation 119883 = (119905
119894 x119894)119873
119894=0 119886 = 119905
0lt 1199051lt sdot sdot sdot lt 119905
119873= 119887
119873 ≪ 119870 Assume that119883 sub 119885 We fit matrix parametersD119899to
minimize functional119870
sum
119896=0
1003816100381610038161003816w119896 minus g⋆ (119911119896)1003816100381610038161003816
2
(26)
It is necessary to use results of previous section Approxi-mate 119885 by constant piecewise vector-function g [119886 119887] rarr
R119872 More precisely g(119911) = w119896 where (119911
119896w119896) isin 119885 119911
119896
is the nearest approximation neighbor of 119911 By substitutingintegrals in (22) to discretization points sums we obtain
D119899
= ( sum
119911119896isin[119905119899minus1119905119899]
(g (119911119896) minus u119899(119911119896)) (g ∘ 119908
119899(119911119896) minus k119899(119911119896))119879
)
sdot ( sum
119911119896isin[119905119899minus1119905119899]
(g ∘ 119908119899(119911119896) minus k119899(119911119896))
Abstract and Applied Analysis 5
64
5686
4972
4258
3544
2829
2115
1401
6871
minus02697
minus7411minus1 06 22 38 54 7 86 102 118 134 15
(a)
minus1 06 22 38 54 7 86 102 117 134 15
64
5736
5072
4408
3745
3081
2417
1753
1089
4253
minus2385
(b)
Figure 3 Approximation of vector-function g(119905) = (119905(119905 minus 2) (119905 minus 1)2(119905 + 1)
2) by fractal interpolation function g⋆ with three (a) and four (b)
points of interpolation correspondingly
sdot (g ∘ 119908119899(119911119896) minus k119899(119911119896))119879
)
minus1
119899 = 1119873
(27)
It is sufficient to apply (1) for constructing fractal interpola-tion vector-function after we findD
119899
Consider several examples of approximation of discretedata
Example 6 Let us approximate vector-function g(119905) = (119905(119905 minus
2) (119905 minus 1)2(119905 + 1)
2) where 119905 isin [minus3 3] Figure 3 shows the
results Here we have two pictures the first one illustratesinitial vector-function and its approximation with 3 pointsand the second one with 4 points where two functions arenearly identical
In this case affine transformations (1) have the followingform
1198601(
119905
1199091
1199092
) = (
05 0 0
minus08842 00943 minus01045
minus01038 minus00530 02287
)(
119905
1199091
1199092
)
+(
0
07320
53989
)
1198602(
119905
1199091
1199092
) = (
05 0 0
34602 07065 minus15549
minus04554 01504 minus02847
)(
119905
1199091
1199092
)
+(
075
108875
81844
)
(28)
Remark 7 Vectors c119899in matrices of affine transformations
(1) equal 0 (like in previous example) It means that fractalinterpolation vector-function can be treated as attractor ofclassical affine IFS in R119872
Example 8 Next example is devoted to a circle g(119905) =
(cos 119905 sin 119905) 119905 isin [0 2120587] Figure 4 shows the results Here wealso have two pictures the first one illustrates initial vector-function and its approximation with 3 points and the secondone with 5 points
In this case affine transformations (1) have the followingform
1198601(
119905
1199091
1199092
) = (
05 0 0
minus03180 00006 02128
minus00013 minus05686 minus00038
)(
119905
1199091
1199092
)
+(
0
09993
05686
)
1198602(
119905
1199091
1199092
) = (
05 0 0
03181 minus00053 minus02128
00040 05686 00020
)(
119905
1199091
1199092
)
+(
3151
minus09945
minus05770
)
(29)
Example 9 Spiral of Archimedes g(119905) = (119905 cos 119905 119905 sin 119905) 119905 isin[0 5120587] where the scheme is equal to the examples above buthere we use far more points of interpolation as illustrated inFigure 5
6 Abstract and Applied Analysis
1
08
06
04
02
111e minus 05
minus02
minus04
minus06
minus08
minus1
minus1133
minus09065
minus06798
minus04532
minus02266
4005eminus05
02267
04533
06799
09055
1133
(a)
1
08
06
04
02
111e minus 06
minus02
minus04
minus06
minus08
minus1
minus1
minus08
minus06
minus04
minus22
minus1099eminus06
02
04
06
08 1
(b)
Figure 4 Approximation of vector-function g(119905) = (cos 119905 sin 119905) by fractal interpolation function g⋆ with three (a) and five (b) points ofinterpolation correspondingly
143
1116
8014
4872
173
minus1412
minus4554
minus7696
minus1084
minus1398
minus1712
minus1251
minus9668
minus6831
minus3995
minus1158
1679
4516
7353
1019
1303
1586
(a)
1261
9775
6944
4113
1283
minus1548
minus4379
minus721
minus1004
minus1287
minus157
minus1152
minus8856
minus6191
minus3526
minus08612
1804
4469
7134
9798
1246
15
(b)
Figure 5 Approximation of vector-function g(119905) = (119905 cos 119905 119905 sin 119905) 119905 isin [0 5120587] by fractal interpolation function g⋆ with twelve (a) andseventeen (b) points of interpolation correspondingly
Example 10 Figure 6 shows approximation of vector-function g(119905) = (cos(15119905) sin(119905)) 119905 isin [0 12120587] byfractal interpolation vector-function with sixteen pointsof interpolation
Example 11 The example illustrates approximation of graphof Weierstrass function 120596(119909) = sum
infin
119899=0(12)119899 cos(21205874119899119909)
(Figure 7) by fractal interpolation vector-function
This example is taken from [10] where fractal approxima-tion is used for approximate calculation of box dimension offractal curves
5 Conclusion
In this paper we have introduced new effective method ofapproximation of continuous vector-functions and vector
Abstract and Applied Analysis 7
1021
08172
06132
04092
02051
0001096
minus02029
minus0407
minus0611
minus0815
minus1019
minus1025
minus08177
minus06109
minus0404
minus01972
0009691
02165
04234
06303
08371
1044
Figure 6 Approximation of vector-function g(119905) = (cos(15119905)sin(119905)) by fractal interpolation function g⋆
2
15
1
05
0
minus05
minus1
minus15
minus2
minus1 minus00 minus0e minus04 minus32 0 02 04 06 30 1
Figure 7 Weierstrass function (blue one) and approximatingvector-function (red one)
sequences by fractal interpolation vector-functions whichare affine transformations withmatrix parameters Parameterfitting was a crucial part of approximation process We havefound appropriate parameter values of fractal interpolationvector-functions and illustrate it with several examples ofdifferent types of discrete data
We assume that fractal approximation is highly promisingcomputational tool for different types of data and it canbe used in many ways even in interdisciplinary fieldswith a quite high precision that allows us to apply fractalapproximation methods to a wide variety of curves smoothand nonsmooth alike
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The work was performed according to the Russian Gov-ernment Program of Competitive Growth of Kazan Federal
University The authors greatly acknowledge the anonymousreviewer for carefully reading the paper and providing con-structive comments that have led to an improved paper
References
[1] M F Barnsley ldquoFractal functions and interpolationrdquo Construc-tive Approximation vol 2 no 4 pp 303ndash329 1986
[2] M F Barnsley J Elton D Hardin and P Massopust ldquoHiddenvariable fractal interpolation functionsrdquo SIAM Journal onMath-ematical Analysis vol 20 no 5 pp 1218ndash1242 1989
[3] M F Barnsley and A N Harrington ldquoThe calculus of fractalinterpolation functionsrdquo Journal of Approximation Theory vol57 no 1 pp 14ndash34 1989
[4] P Massopust Interpolation and Approximation with Splines andFractals Oxford University Press Oxford UK 2010
[5] K Igudesman and G Shabernev ldquoNovel method of fractalapproximationrdquo Lobachevskii Journal of Mathematics vol 34no 2 pp 125ndash132 2013
[6] J E Hutchinson ldquoFractals and self-similarityrdquo Indiana Univer-sity Mathematics Journal vol 30 no 5 pp 713ndash747 1981
[7] M F Barnsley and L P Hurd Fractal Image Compression A KPeters Wellesley Mass USA 1993
[8] M Barnsley Fractals Everywhere Academic Press BostonMass USA 1988
[9] J R Magnus and H Neudecker ldquoMatrix differential calculuswith applications in statistics and econometricsrdquoApplied Math-ematical Sciences vol 8 no 144 pp 7175ndash7181 2014
[10] K Igudesman R Lavrenov and V Klassen Novel Approach toCalculation of Box Dimension of Fractal Functions Wiley 3rdedition 2007
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 3
10
05
minus05
minus10
02 04 06 08 10
Figure 1 Fractal interpolation vector-function g⋆
Thus Φ 119871119872
2[119886 119887] rarr 119871
119872
2[119886 119887] is a contractive operator and
g⋆ is its fixed pointInstead of minimizing g minus g⋆
2we minimize g minus Φg
2
that makes the problem of optimization much easier Thecollage theorem provides validity of such approach [8]
Theorem 3 Let (119883 119889) be complete metric space and 119879 119883 rarr
119883 is contractive mapping with contraction coefficient 119888 isin [0 1)and fixed point 119909⋆ Then
119889 (119909 119909⋆) le
119889 (119909 119879 (119909))
1 minus 119888(13)
for all 119909 isin 119883
Considering (4) and (6) rewrite (7)
(Φg) (119905) =119873
sum
119899=1
(u119899(119905) +D
119899(g ∘ 119908
119899(119905) minus k
119899(119905))) 120594
[119905119899minus1119905119899](119905)
(14)
where
u119899(119905) =
(x119899minus x119899minus1
) 119905 + (119905119899x119899minus1
minus 119905119899minus1
x119899)
119905119899minus 119905119899minus1
k119899(119905) =
(x119873minus x0) 119905 + (119905
119899x0minus 119905119899minus1
x119873)
119905119899minus 119905119899minus1
119908119899(119905) =
(119887 minus 119886) 119905 + (119905119899119886 minus 119905119899minus1
119887)
119905119899minus 119905119899minus1
(15)
Thus we minimize the functional
1003817100381710038171003817g minus Φg1003817100381710038171003817
2
2
=
119873
sum
119899=1
int
119905119899
119905119899minus1
1003816100381610038161003816g (119905) minus u119899(119905) minusD
119899(g ∘ 119908
119899(119905) minus k
119899(119905))
1003816100381610038161003816
2 d119905(16)
Lemma 4 Let f h isin 119871119872
2[119886 119887] be square integrated vector-
functions Suppose that matrix int119887
119886hh119879dt is nondegenerated
Matrix integration is implied to be componentwise Then thefunctional
Ψ R119872times119872
997888rarr R Ψ (X) = int119887
119886
|f minus Xh|2 dt (17)
reaches its minimum in X = int119887
119886fh119879dt (intba hhTdt)minus1
Proof To prove it we use matrix differential calculus [9]Consider
dΨ (XU) = d(int119887
119886
(f minus Xh)119879 (f minus Xh) d119905)U
= d(int119887
119886
(f119879f minus h119879X119879f minus f119879Xh + h119879X119879Xh) d119905)U
= int
119887
119886
(minush119879U119879f minus f119879Uh + h119879U119879Xh + h119879X119879Uh) d119905
= 2int
119887
119886
(minush119879U119879f + h119879U119879Xh) d119905(18)
Necessary condition of existence of functional Ψ extremumis dΨ(XU) = 0 for all U isin R119872times119872 Since there is 119880-linearityof functional dΨ(XU) it is sufficient to prove dΨ(XU) = 0
only formatricesU that consist of1198722minus1 zeros and one unityTherefore we have1198722 expressions for finding coefficients ofmatrix X In matrix form these expressions are as follows
int
119887
119886
fh119879d119905 = int119887
119886
Xhh119879d119905 (19)
from which
X = int
119887
119886
fh119879d119905 (int119887
119886
hh119879d119905)minus1
(20)
Hence
d2Ψ (XU) = 2int119887
119886
h119879U119879Uh d119905 = 2int119887
119886
|Uh|2 d119905 ge 0 (21)
and then functional Ψ is convex one Thus the value X isabsolute minimum of Ψ
From Lemma 4 it follows that functional (16) reachesminimum when
D119899= int
119905119899
119905119899minus1
(g (119905) minus u119899(119905)) (g ∘ 119908
119899(119905) minus k
119899(119905))119879 d119905
sdot (int
119905119899
119905119899minus1
(g ∘ 119908119899(119905) minus k
119899(119905)) (g ∘ 119908
119899(119905) minus k
119899(119905))119879 d119905)minus1
(22)
4 Abstract and Applied Analysis
1
09
08
07
06
05
04
03
02
01
7644e minus 31
minus1
minus08
minus06
minus04
minus02
02
04
06
08 1
2165eminus15
Figure 2 Vector-function g(119905) = (1199052 1199053) and fractal interpolation
vector-function g⋆ completely identical
Example 5 Let us approximate vector-function g(119905) =
(1199052 1199053) on segment [minus1 1] by the fractal interpolation vector-
function constructed on values of g(119905) in points 1199050= minus1 119905
1=
0 and 1199052= 1 and 119909
0= (1 minus1) 119909
1= (0 0) and 119909
2= (1 1)
(see Figure 2) Then
1198861= 1198862=1
2
1198901= minus
1
2 119890
2=1
2
u1= (minus119905 119905) u
2= (119905 119905)
k1= (1 1 + 2119905) k
2= (1 minus1 + 2119905)
1199081= 1 + 2119905 119908
2= minus1 + 2119905
2
(23)
CalculateD1D2according to formula (22) as follows
D1= (
1
40
minus3
8
1
8
) D2= (
1
40
3
8
1
8
) (24)
Apply affine transformations from (1) to vector 119905 1199052 1199053
1198601(
119905
1199052
1199053
) =(
(
1
20 0
minus1
2
1
40
3
8minus3
8
1
8
)
)
(
119905
1199052
1199053
)+(
(
minus1
2
1
4
minus1
8
)
)
=((
(
119905
2minus1
2
1199052
4minus119905
2+1
4
1199053
8minus31199052
8+3119905
8minus1
8
))
)
=((
(
119905minus 1
2
(119905 minus 1
2)
2
(119905 minus 1
2)
3
))
)
1198602(
119905
1199052
1199053
) =(
(
1
20 0
1
2
1
40
3
8
3
8
1
8
)
)
(
119905
1199052
1199053
)+(
(
1
2
1
4
1
8
)
)
=((
(
119905
2+1
2
1199052
4+119905
2+1
4
1199053
8+31199052
8+3119905
8+1
8
))
)
=((
(
119905+ 1
2
(119905 + 1
2)
2
(119905 + 1
2)
3
))
)
(25)
Thus Φ(g) = g and g = g⋆
4 Discretization and Results
In this section we approximate discrete data 119885 =
(119911119898w119898)119870
119896=0 119886 = 119911
0lt 1199111lt sdot sdot sdot lt 119911
119870= 119887 by fractal
interpolation vector-function g⋆ constructed on points ofinterpolation 119883 = (119905
119894 x119894)119873
119894=0 119886 = 119905
0lt 1199051lt sdot sdot sdot lt 119905
119873= 119887
119873 ≪ 119870 Assume that119883 sub 119885 We fit matrix parametersD119899to
minimize functional119870
sum
119896=0
1003816100381610038161003816w119896 minus g⋆ (119911119896)1003816100381610038161003816
2
(26)
It is necessary to use results of previous section Approxi-mate 119885 by constant piecewise vector-function g [119886 119887] rarr
R119872 More precisely g(119911) = w119896 where (119911
119896w119896) isin 119885 119911
119896
is the nearest approximation neighbor of 119911 By substitutingintegrals in (22) to discretization points sums we obtain
D119899
= ( sum
119911119896isin[119905119899minus1119905119899]
(g (119911119896) minus u119899(119911119896)) (g ∘ 119908
119899(119911119896) minus k119899(119911119896))119879
)
sdot ( sum
119911119896isin[119905119899minus1119905119899]
(g ∘ 119908119899(119911119896) minus k119899(119911119896))
Abstract and Applied Analysis 5
64
5686
4972
4258
3544
2829
2115
1401
6871
minus02697
minus7411minus1 06 22 38 54 7 86 102 118 134 15
(a)
minus1 06 22 38 54 7 86 102 117 134 15
64
5736
5072
4408
3745
3081
2417
1753
1089
4253
minus2385
(b)
Figure 3 Approximation of vector-function g(119905) = (119905(119905 minus 2) (119905 minus 1)2(119905 + 1)
2) by fractal interpolation function g⋆ with three (a) and four (b)
points of interpolation correspondingly
sdot (g ∘ 119908119899(119911119896) minus k119899(119911119896))119879
)
minus1
119899 = 1119873
(27)
It is sufficient to apply (1) for constructing fractal interpola-tion vector-function after we findD
119899
Consider several examples of approximation of discretedata
Example 6 Let us approximate vector-function g(119905) = (119905(119905 minus
2) (119905 minus 1)2(119905 + 1)
2) where 119905 isin [minus3 3] Figure 3 shows the
results Here we have two pictures the first one illustratesinitial vector-function and its approximation with 3 pointsand the second one with 4 points where two functions arenearly identical
In this case affine transformations (1) have the followingform
1198601(
119905
1199091
1199092
) = (
05 0 0
minus08842 00943 minus01045
minus01038 minus00530 02287
)(
119905
1199091
1199092
)
+(
0
07320
53989
)
1198602(
119905
1199091
1199092
) = (
05 0 0
34602 07065 minus15549
minus04554 01504 minus02847
)(
119905
1199091
1199092
)
+(
075
108875
81844
)
(28)
Remark 7 Vectors c119899in matrices of affine transformations
(1) equal 0 (like in previous example) It means that fractalinterpolation vector-function can be treated as attractor ofclassical affine IFS in R119872
Example 8 Next example is devoted to a circle g(119905) =
(cos 119905 sin 119905) 119905 isin [0 2120587] Figure 4 shows the results Here wealso have two pictures the first one illustrates initial vector-function and its approximation with 3 points and the secondone with 5 points
In this case affine transformations (1) have the followingform
1198601(
119905
1199091
1199092
) = (
05 0 0
minus03180 00006 02128
minus00013 minus05686 minus00038
)(
119905
1199091
1199092
)
+(
0
09993
05686
)
1198602(
119905
1199091
1199092
) = (
05 0 0
03181 minus00053 minus02128
00040 05686 00020
)(
119905
1199091
1199092
)
+(
3151
minus09945
minus05770
)
(29)
Example 9 Spiral of Archimedes g(119905) = (119905 cos 119905 119905 sin 119905) 119905 isin[0 5120587] where the scheme is equal to the examples above buthere we use far more points of interpolation as illustrated inFigure 5
6 Abstract and Applied Analysis
1
08
06
04
02
111e minus 05
minus02
minus04
minus06
minus08
minus1
minus1133
minus09065
minus06798
minus04532
minus02266
4005eminus05
02267
04533
06799
09055
1133
(a)
1
08
06
04
02
111e minus 06
minus02
minus04
minus06
minus08
minus1
minus1
minus08
minus06
minus04
minus22
minus1099eminus06
02
04
06
08 1
(b)
Figure 4 Approximation of vector-function g(119905) = (cos 119905 sin 119905) by fractal interpolation function g⋆ with three (a) and five (b) points ofinterpolation correspondingly
143
1116
8014
4872
173
minus1412
minus4554
minus7696
minus1084
minus1398
minus1712
minus1251
minus9668
minus6831
minus3995
minus1158
1679
4516
7353
1019
1303
1586
(a)
1261
9775
6944
4113
1283
minus1548
minus4379
minus721
minus1004
minus1287
minus157
minus1152
minus8856
minus6191
minus3526
minus08612
1804
4469
7134
9798
1246
15
(b)
Figure 5 Approximation of vector-function g(119905) = (119905 cos 119905 119905 sin 119905) 119905 isin [0 5120587] by fractal interpolation function g⋆ with twelve (a) andseventeen (b) points of interpolation correspondingly
Example 10 Figure 6 shows approximation of vector-function g(119905) = (cos(15119905) sin(119905)) 119905 isin [0 12120587] byfractal interpolation vector-function with sixteen pointsof interpolation
Example 11 The example illustrates approximation of graphof Weierstrass function 120596(119909) = sum
infin
119899=0(12)119899 cos(21205874119899119909)
(Figure 7) by fractal interpolation vector-function
This example is taken from [10] where fractal approxima-tion is used for approximate calculation of box dimension offractal curves
5 Conclusion
In this paper we have introduced new effective method ofapproximation of continuous vector-functions and vector
Abstract and Applied Analysis 7
1021
08172
06132
04092
02051
0001096
minus02029
minus0407
minus0611
minus0815
minus1019
minus1025
minus08177
minus06109
minus0404
minus01972
0009691
02165
04234
06303
08371
1044
Figure 6 Approximation of vector-function g(119905) = (cos(15119905)sin(119905)) by fractal interpolation function g⋆
2
15
1
05
0
minus05
minus1
minus15
minus2
minus1 minus00 minus0e minus04 minus32 0 02 04 06 30 1
Figure 7 Weierstrass function (blue one) and approximatingvector-function (red one)
sequences by fractal interpolation vector-functions whichare affine transformations withmatrix parameters Parameterfitting was a crucial part of approximation process We havefound appropriate parameter values of fractal interpolationvector-functions and illustrate it with several examples ofdifferent types of discrete data
We assume that fractal approximation is highly promisingcomputational tool for different types of data and it canbe used in many ways even in interdisciplinary fieldswith a quite high precision that allows us to apply fractalapproximation methods to a wide variety of curves smoothand nonsmooth alike
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The work was performed according to the Russian Gov-ernment Program of Competitive Growth of Kazan Federal
University The authors greatly acknowledge the anonymousreviewer for carefully reading the paper and providing con-structive comments that have led to an improved paper
References
[1] M F Barnsley ldquoFractal functions and interpolationrdquo Construc-tive Approximation vol 2 no 4 pp 303ndash329 1986
[2] M F Barnsley J Elton D Hardin and P Massopust ldquoHiddenvariable fractal interpolation functionsrdquo SIAM Journal onMath-ematical Analysis vol 20 no 5 pp 1218ndash1242 1989
[3] M F Barnsley and A N Harrington ldquoThe calculus of fractalinterpolation functionsrdquo Journal of Approximation Theory vol57 no 1 pp 14ndash34 1989
[4] P Massopust Interpolation and Approximation with Splines andFractals Oxford University Press Oxford UK 2010
[5] K Igudesman and G Shabernev ldquoNovel method of fractalapproximationrdquo Lobachevskii Journal of Mathematics vol 34no 2 pp 125ndash132 2013
[6] J E Hutchinson ldquoFractals and self-similarityrdquo Indiana Univer-sity Mathematics Journal vol 30 no 5 pp 713ndash747 1981
[7] M F Barnsley and L P Hurd Fractal Image Compression A KPeters Wellesley Mass USA 1993
[8] M Barnsley Fractals Everywhere Academic Press BostonMass USA 1988
[9] J R Magnus and H Neudecker ldquoMatrix differential calculuswith applications in statistics and econometricsrdquoApplied Math-ematical Sciences vol 8 no 144 pp 7175ndash7181 2014
[10] K Igudesman R Lavrenov and V Klassen Novel Approach toCalculation of Box Dimension of Fractal Functions Wiley 3rdedition 2007
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Abstract and Applied Analysis
1
09
08
07
06
05
04
03
02
01
7644e minus 31
minus1
minus08
minus06
minus04
minus02
02
04
06
08 1
2165eminus15
Figure 2 Vector-function g(119905) = (1199052 1199053) and fractal interpolation
vector-function g⋆ completely identical
Example 5 Let us approximate vector-function g(119905) =
(1199052 1199053) on segment [minus1 1] by the fractal interpolation vector-
function constructed on values of g(119905) in points 1199050= minus1 119905
1=
0 and 1199052= 1 and 119909
0= (1 minus1) 119909
1= (0 0) and 119909
2= (1 1)
(see Figure 2) Then
1198861= 1198862=1
2
1198901= minus
1
2 119890
2=1
2
u1= (minus119905 119905) u
2= (119905 119905)
k1= (1 1 + 2119905) k
2= (1 minus1 + 2119905)
1199081= 1 + 2119905 119908
2= minus1 + 2119905
2
(23)
CalculateD1D2according to formula (22) as follows
D1= (
1
40
minus3
8
1
8
) D2= (
1
40
3
8
1
8
) (24)
Apply affine transformations from (1) to vector 119905 1199052 1199053
1198601(
119905
1199052
1199053
) =(
(
1
20 0
minus1
2
1
40
3
8minus3
8
1
8
)
)
(
119905
1199052
1199053
)+(
(
minus1
2
1
4
minus1
8
)
)
=((
(
119905
2minus1
2
1199052
4minus119905
2+1
4
1199053
8minus31199052
8+3119905
8minus1
8
))
)
=((
(
119905minus 1
2
(119905 minus 1
2)
2
(119905 minus 1
2)
3
))
)
1198602(
119905
1199052
1199053
) =(
(
1
20 0
1
2
1
40
3
8
3
8
1
8
)
)
(
119905
1199052
1199053
)+(
(
1
2
1
4
1
8
)
)
=((
(
119905
2+1
2
1199052
4+119905
2+1
4
1199053
8+31199052
8+3119905
8+1
8
))
)
=((
(
119905+ 1
2
(119905 + 1
2)
2
(119905 + 1
2)
3
))
)
(25)
Thus Φ(g) = g and g = g⋆
4 Discretization and Results
In this section we approximate discrete data 119885 =
(119911119898w119898)119870
119896=0 119886 = 119911
0lt 1199111lt sdot sdot sdot lt 119911
119870= 119887 by fractal
interpolation vector-function g⋆ constructed on points ofinterpolation 119883 = (119905
119894 x119894)119873
119894=0 119886 = 119905
0lt 1199051lt sdot sdot sdot lt 119905
119873= 119887
119873 ≪ 119870 Assume that119883 sub 119885 We fit matrix parametersD119899to
minimize functional119870
sum
119896=0
1003816100381610038161003816w119896 minus g⋆ (119911119896)1003816100381610038161003816
2
(26)
It is necessary to use results of previous section Approxi-mate 119885 by constant piecewise vector-function g [119886 119887] rarr
R119872 More precisely g(119911) = w119896 where (119911
119896w119896) isin 119885 119911
119896
is the nearest approximation neighbor of 119911 By substitutingintegrals in (22) to discretization points sums we obtain
D119899
= ( sum
119911119896isin[119905119899minus1119905119899]
(g (119911119896) minus u119899(119911119896)) (g ∘ 119908
119899(119911119896) minus k119899(119911119896))119879
)
sdot ( sum
119911119896isin[119905119899minus1119905119899]
(g ∘ 119908119899(119911119896) minus k119899(119911119896))
Abstract and Applied Analysis 5
64
5686
4972
4258
3544
2829
2115
1401
6871
minus02697
minus7411minus1 06 22 38 54 7 86 102 118 134 15
(a)
minus1 06 22 38 54 7 86 102 117 134 15
64
5736
5072
4408
3745
3081
2417
1753
1089
4253
minus2385
(b)
Figure 3 Approximation of vector-function g(119905) = (119905(119905 minus 2) (119905 minus 1)2(119905 + 1)
2) by fractal interpolation function g⋆ with three (a) and four (b)
points of interpolation correspondingly
sdot (g ∘ 119908119899(119911119896) minus k119899(119911119896))119879
)
minus1
119899 = 1119873
(27)
It is sufficient to apply (1) for constructing fractal interpola-tion vector-function after we findD
119899
Consider several examples of approximation of discretedata
Example 6 Let us approximate vector-function g(119905) = (119905(119905 minus
2) (119905 minus 1)2(119905 + 1)
2) where 119905 isin [minus3 3] Figure 3 shows the
results Here we have two pictures the first one illustratesinitial vector-function and its approximation with 3 pointsand the second one with 4 points where two functions arenearly identical
In this case affine transformations (1) have the followingform
1198601(
119905
1199091
1199092
) = (
05 0 0
minus08842 00943 minus01045
minus01038 minus00530 02287
)(
119905
1199091
1199092
)
+(
0
07320
53989
)
1198602(
119905
1199091
1199092
) = (
05 0 0
34602 07065 minus15549
minus04554 01504 minus02847
)(
119905
1199091
1199092
)
+(
075
108875
81844
)
(28)
Remark 7 Vectors c119899in matrices of affine transformations
(1) equal 0 (like in previous example) It means that fractalinterpolation vector-function can be treated as attractor ofclassical affine IFS in R119872
Example 8 Next example is devoted to a circle g(119905) =
(cos 119905 sin 119905) 119905 isin [0 2120587] Figure 4 shows the results Here wealso have two pictures the first one illustrates initial vector-function and its approximation with 3 points and the secondone with 5 points
In this case affine transformations (1) have the followingform
1198601(
119905
1199091
1199092
) = (
05 0 0
minus03180 00006 02128
minus00013 minus05686 minus00038
)(
119905
1199091
1199092
)
+(
0
09993
05686
)
1198602(
119905
1199091
1199092
) = (
05 0 0
03181 minus00053 minus02128
00040 05686 00020
)(
119905
1199091
1199092
)
+(
3151
minus09945
minus05770
)
(29)
Example 9 Spiral of Archimedes g(119905) = (119905 cos 119905 119905 sin 119905) 119905 isin[0 5120587] where the scheme is equal to the examples above buthere we use far more points of interpolation as illustrated inFigure 5
6 Abstract and Applied Analysis
1
08
06
04
02
111e minus 05
minus02
minus04
minus06
minus08
minus1
minus1133
minus09065
minus06798
minus04532
minus02266
4005eminus05
02267
04533
06799
09055
1133
(a)
1
08
06
04
02
111e minus 06
minus02
minus04
minus06
minus08
minus1
minus1
minus08
minus06
minus04
minus22
minus1099eminus06
02
04
06
08 1
(b)
Figure 4 Approximation of vector-function g(119905) = (cos 119905 sin 119905) by fractal interpolation function g⋆ with three (a) and five (b) points ofinterpolation correspondingly
143
1116
8014
4872
173
minus1412
minus4554
minus7696
minus1084
minus1398
minus1712
minus1251
minus9668
minus6831
minus3995
minus1158
1679
4516
7353
1019
1303
1586
(a)
1261
9775
6944
4113
1283
minus1548
minus4379
minus721
minus1004
minus1287
minus157
minus1152
minus8856
minus6191
minus3526
minus08612
1804
4469
7134
9798
1246
15
(b)
Figure 5 Approximation of vector-function g(119905) = (119905 cos 119905 119905 sin 119905) 119905 isin [0 5120587] by fractal interpolation function g⋆ with twelve (a) andseventeen (b) points of interpolation correspondingly
Example 10 Figure 6 shows approximation of vector-function g(119905) = (cos(15119905) sin(119905)) 119905 isin [0 12120587] byfractal interpolation vector-function with sixteen pointsof interpolation
Example 11 The example illustrates approximation of graphof Weierstrass function 120596(119909) = sum
infin
119899=0(12)119899 cos(21205874119899119909)
(Figure 7) by fractal interpolation vector-function
This example is taken from [10] where fractal approxima-tion is used for approximate calculation of box dimension offractal curves
5 Conclusion
In this paper we have introduced new effective method ofapproximation of continuous vector-functions and vector
Abstract and Applied Analysis 7
1021
08172
06132
04092
02051
0001096
minus02029
minus0407
minus0611
minus0815
minus1019
minus1025
minus08177
minus06109
minus0404
minus01972
0009691
02165
04234
06303
08371
1044
Figure 6 Approximation of vector-function g(119905) = (cos(15119905)sin(119905)) by fractal interpolation function g⋆
2
15
1
05
0
minus05
minus1
minus15
minus2
minus1 minus00 minus0e minus04 minus32 0 02 04 06 30 1
Figure 7 Weierstrass function (blue one) and approximatingvector-function (red one)
sequences by fractal interpolation vector-functions whichare affine transformations withmatrix parameters Parameterfitting was a crucial part of approximation process We havefound appropriate parameter values of fractal interpolationvector-functions and illustrate it with several examples ofdifferent types of discrete data
We assume that fractal approximation is highly promisingcomputational tool for different types of data and it canbe used in many ways even in interdisciplinary fieldswith a quite high precision that allows us to apply fractalapproximation methods to a wide variety of curves smoothand nonsmooth alike
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The work was performed according to the Russian Gov-ernment Program of Competitive Growth of Kazan Federal
University The authors greatly acknowledge the anonymousreviewer for carefully reading the paper and providing con-structive comments that have led to an improved paper
References
[1] M F Barnsley ldquoFractal functions and interpolationrdquo Construc-tive Approximation vol 2 no 4 pp 303ndash329 1986
[2] M F Barnsley J Elton D Hardin and P Massopust ldquoHiddenvariable fractal interpolation functionsrdquo SIAM Journal onMath-ematical Analysis vol 20 no 5 pp 1218ndash1242 1989
[3] M F Barnsley and A N Harrington ldquoThe calculus of fractalinterpolation functionsrdquo Journal of Approximation Theory vol57 no 1 pp 14ndash34 1989
[4] P Massopust Interpolation and Approximation with Splines andFractals Oxford University Press Oxford UK 2010
[5] K Igudesman and G Shabernev ldquoNovel method of fractalapproximationrdquo Lobachevskii Journal of Mathematics vol 34no 2 pp 125ndash132 2013
[6] J E Hutchinson ldquoFractals and self-similarityrdquo Indiana Univer-sity Mathematics Journal vol 30 no 5 pp 713ndash747 1981
[7] M F Barnsley and L P Hurd Fractal Image Compression A KPeters Wellesley Mass USA 1993
[8] M Barnsley Fractals Everywhere Academic Press BostonMass USA 1988
[9] J R Magnus and H Neudecker ldquoMatrix differential calculuswith applications in statistics and econometricsrdquoApplied Math-ematical Sciences vol 8 no 144 pp 7175ndash7181 2014
[10] K Igudesman R Lavrenov and V Klassen Novel Approach toCalculation of Box Dimension of Fractal Functions Wiley 3rdedition 2007
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 5
64
5686
4972
4258
3544
2829
2115
1401
6871
minus02697
minus7411minus1 06 22 38 54 7 86 102 118 134 15
(a)
minus1 06 22 38 54 7 86 102 117 134 15
64
5736
5072
4408
3745
3081
2417
1753
1089
4253
minus2385
(b)
Figure 3 Approximation of vector-function g(119905) = (119905(119905 minus 2) (119905 minus 1)2(119905 + 1)
2) by fractal interpolation function g⋆ with three (a) and four (b)
points of interpolation correspondingly
sdot (g ∘ 119908119899(119911119896) minus k119899(119911119896))119879
)
minus1
119899 = 1119873
(27)
It is sufficient to apply (1) for constructing fractal interpola-tion vector-function after we findD
119899
Consider several examples of approximation of discretedata
Example 6 Let us approximate vector-function g(119905) = (119905(119905 minus
2) (119905 minus 1)2(119905 + 1)
2) where 119905 isin [minus3 3] Figure 3 shows the
results Here we have two pictures the first one illustratesinitial vector-function and its approximation with 3 pointsand the second one with 4 points where two functions arenearly identical
In this case affine transformations (1) have the followingform
1198601(
119905
1199091
1199092
) = (
05 0 0
minus08842 00943 minus01045
minus01038 minus00530 02287
)(
119905
1199091
1199092
)
+(
0
07320
53989
)
1198602(
119905
1199091
1199092
) = (
05 0 0
34602 07065 minus15549
minus04554 01504 minus02847
)(
119905
1199091
1199092
)
+(
075
108875
81844
)
(28)
Remark 7 Vectors c119899in matrices of affine transformations
(1) equal 0 (like in previous example) It means that fractalinterpolation vector-function can be treated as attractor ofclassical affine IFS in R119872
Example 8 Next example is devoted to a circle g(119905) =
(cos 119905 sin 119905) 119905 isin [0 2120587] Figure 4 shows the results Here wealso have two pictures the first one illustrates initial vector-function and its approximation with 3 points and the secondone with 5 points
In this case affine transformations (1) have the followingform
1198601(
119905
1199091
1199092
) = (
05 0 0
minus03180 00006 02128
minus00013 minus05686 minus00038
)(
119905
1199091
1199092
)
+(
0
09993
05686
)
1198602(
119905
1199091
1199092
) = (
05 0 0
03181 minus00053 minus02128
00040 05686 00020
)(
119905
1199091
1199092
)
+(
3151
minus09945
minus05770
)
(29)
Example 9 Spiral of Archimedes g(119905) = (119905 cos 119905 119905 sin 119905) 119905 isin[0 5120587] where the scheme is equal to the examples above buthere we use far more points of interpolation as illustrated inFigure 5
6 Abstract and Applied Analysis
1
08
06
04
02
111e minus 05
minus02
minus04
minus06
minus08
minus1
minus1133
minus09065
minus06798
minus04532
minus02266
4005eminus05
02267
04533
06799
09055
1133
(a)
1
08
06
04
02
111e minus 06
minus02
minus04
minus06
minus08
minus1
minus1
minus08
minus06
minus04
minus22
minus1099eminus06
02
04
06
08 1
(b)
Figure 4 Approximation of vector-function g(119905) = (cos 119905 sin 119905) by fractal interpolation function g⋆ with three (a) and five (b) points ofinterpolation correspondingly
143
1116
8014
4872
173
minus1412
minus4554
minus7696
minus1084
minus1398
minus1712
minus1251
minus9668
minus6831
minus3995
minus1158
1679
4516
7353
1019
1303
1586
(a)
1261
9775
6944
4113
1283
minus1548
minus4379
minus721
minus1004
minus1287
minus157
minus1152
minus8856
minus6191
minus3526
minus08612
1804
4469
7134
9798
1246
15
(b)
Figure 5 Approximation of vector-function g(119905) = (119905 cos 119905 119905 sin 119905) 119905 isin [0 5120587] by fractal interpolation function g⋆ with twelve (a) andseventeen (b) points of interpolation correspondingly
Example 10 Figure 6 shows approximation of vector-function g(119905) = (cos(15119905) sin(119905)) 119905 isin [0 12120587] byfractal interpolation vector-function with sixteen pointsof interpolation
Example 11 The example illustrates approximation of graphof Weierstrass function 120596(119909) = sum
infin
119899=0(12)119899 cos(21205874119899119909)
(Figure 7) by fractal interpolation vector-function
This example is taken from [10] where fractal approxima-tion is used for approximate calculation of box dimension offractal curves
5 Conclusion
In this paper we have introduced new effective method ofapproximation of continuous vector-functions and vector
Abstract and Applied Analysis 7
1021
08172
06132
04092
02051
0001096
minus02029
minus0407
minus0611
minus0815
minus1019
minus1025
minus08177
minus06109
minus0404
minus01972
0009691
02165
04234
06303
08371
1044
Figure 6 Approximation of vector-function g(119905) = (cos(15119905)sin(119905)) by fractal interpolation function g⋆
2
15
1
05
0
minus05
minus1
minus15
minus2
minus1 minus00 minus0e minus04 minus32 0 02 04 06 30 1
Figure 7 Weierstrass function (blue one) and approximatingvector-function (red one)
sequences by fractal interpolation vector-functions whichare affine transformations withmatrix parameters Parameterfitting was a crucial part of approximation process We havefound appropriate parameter values of fractal interpolationvector-functions and illustrate it with several examples ofdifferent types of discrete data
We assume that fractal approximation is highly promisingcomputational tool for different types of data and it canbe used in many ways even in interdisciplinary fieldswith a quite high precision that allows us to apply fractalapproximation methods to a wide variety of curves smoothand nonsmooth alike
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The work was performed according to the Russian Gov-ernment Program of Competitive Growth of Kazan Federal
University The authors greatly acknowledge the anonymousreviewer for carefully reading the paper and providing con-structive comments that have led to an improved paper
References
[1] M F Barnsley ldquoFractal functions and interpolationrdquo Construc-tive Approximation vol 2 no 4 pp 303ndash329 1986
[2] M F Barnsley J Elton D Hardin and P Massopust ldquoHiddenvariable fractal interpolation functionsrdquo SIAM Journal onMath-ematical Analysis vol 20 no 5 pp 1218ndash1242 1989
[3] M F Barnsley and A N Harrington ldquoThe calculus of fractalinterpolation functionsrdquo Journal of Approximation Theory vol57 no 1 pp 14ndash34 1989
[4] P Massopust Interpolation and Approximation with Splines andFractals Oxford University Press Oxford UK 2010
[5] K Igudesman and G Shabernev ldquoNovel method of fractalapproximationrdquo Lobachevskii Journal of Mathematics vol 34no 2 pp 125ndash132 2013
[6] J E Hutchinson ldquoFractals and self-similarityrdquo Indiana Univer-sity Mathematics Journal vol 30 no 5 pp 713ndash747 1981
[7] M F Barnsley and L P Hurd Fractal Image Compression A KPeters Wellesley Mass USA 1993
[8] M Barnsley Fractals Everywhere Academic Press BostonMass USA 1988
[9] J R Magnus and H Neudecker ldquoMatrix differential calculuswith applications in statistics and econometricsrdquoApplied Math-ematical Sciences vol 8 no 144 pp 7175ndash7181 2014
[10] K Igudesman R Lavrenov and V Klassen Novel Approach toCalculation of Box Dimension of Fractal Functions Wiley 3rdedition 2007
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Abstract and Applied Analysis
1
08
06
04
02
111e minus 05
minus02
minus04
minus06
minus08
minus1
minus1133
minus09065
minus06798
minus04532
minus02266
4005eminus05
02267
04533
06799
09055
1133
(a)
1
08
06
04
02
111e minus 06
minus02
minus04
minus06
minus08
minus1
minus1
minus08
minus06
minus04
minus22
minus1099eminus06
02
04
06
08 1
(b)
Figure 4 Approximation of vector-function g(119905) = (cos 119905 sin 119905) by fractal interpolation function g⋆ with three (a) and five (b) points ofinterpolation correspondingly
143
1116
8014
4872
173
minus1412
minus4554
minus7696
minus1084
minus1398
minus1712
minus1251
minus9668
minus6831
minus3995
minus1158
1679
4516
7353
1019
1303
1586
(a)
1261
9775
6944
4113
1283
minus1548
minus4379
minus721
minus1004
minus1287
minus157
minus1152
minus8856
minus6191
minus3526
minus08612
1804
4469
7134
9798
1246
15
(b)
Figure 5 Approximation of vector-function g(119905) = (119905 cos 119905 119905 sin 119905) 119905 isin [0 5120587] by fractal interpolation function g⋆ with twelve (a) andseventeen (b) points of interpolation correspondingly
Example 10 Figure 6 shows approximation of vector-function g(119905) = (cos(15119905) sin(119905)) 119905 isin [0 12120587] byfractal interpolation vector-function with sixteen pointsof interpolation
Example 11 The example illustrates approximation of graphof Weierstrass function 120596(119909) = sum
infin
119899=0(12)119899 cos(21205874119899119909)
(Figure 7) by fractal interpolation vector-function
This example is taken from [10] where fractal approxima-tion is used for approximate calculation of box dimension offractal curves
5 Conclusion
In this paper we have introduced new effective method ofapproximation of continuous vector-functions and vector
Abstract and Applied Analysis 7
1021
08172
06132
04092
02051
0001096
minus02029
minus0407
minus0611
minus0815
minus1019
minus1025
minus08177
minus06109
minus0404
minus01972
0009691
02165
04234
06303
08371
1044
Figure 6 Approximation of vector-function g(119905) = (cos(15119905)sin(119905)) by fractal interpolation function g⋆
2
15
1
05
0
minus05
minus1
minus15
minus2
minus1 minus00 minus0e minus04 minus32 0 02 04 06 30 1
Figure 7 Weierstrass function (blue one) and approximatingvector-function (red one)
sequences by fractal interpolation vector-functions whichare affine transformations withmatrix parameters Parameterfitting was a crucial part of approximation process We havefound appropriate parameter values of fractal interpolationvector-functions and illustrate it with several examples ofdifferent types of discrete data
We assume that fractal approximation is highly promisingcomputational tool for different types of data and it canbe used in many ways even in interdisciplinary fieldswith a quite high precision that allows us to apply fractalapproximation methods to a wide variety of curves smoothand nonsmooth alike
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The work was performed according to the Russian Gov-ernment Program of Competitive Growth of Kazan Federal
University The authors greatly acknowledge the anonymousreviewer for carefully reading the paper and providing con-structive comments that have led to an improved paper
References
[1] M F Barnsley ldquoFractal functions and interpolationrdquo Construc-tive Approximation vol 2 no 4 pp 303ndash329 1986
[2] M F Barnsley J Elton D Hardin and P Massopust ldquoHiddenvariable fractal interpolation functionsrdquo SIAM Journal onMath-ematical Analysis vol 20 no 5 pp 1218ndash1242 1989
[3] M F Barnsley and A N Harrington ldquoThe calculus of fractalinterpolation functionsrdquo Journal of Approximation Theory vol57 no 1 pp 14ndash34 1989
[4] P Massopust Interpolation and Approximation with Splines andFractals Oxford University Press Oxford UK 2010
[5] K Igudesman and G Shabernev ldquoNovel method of fractalapproximationrdquo Lobachevskii Journal of Mathematics vol 34no 2 pp 125ndash132 2013
[6] J E Hutchinson ldquoFractals and self-similarityrdquo Indiana Univer-sity Mathematics Journal vol 30 no 5 pp 713ndash747 1981
[7] M F Barnsley and L P Hurd Fractal Image Compression A KPeters Wellesley Mass USA 1993
[8] M Barnsley Fractals Everywhere Academic Press BostonMass USA 1988
[9] J R Magnus and H Neudecker ldquoMatrix differential calculuswith applications in statistics and econometricsrdquoApplied Math-ematical Sciences vol 8 no 144 pp 7175ndash7181 2014
[10] K Igudesman R Lavrenov and V Klassen Novel Approach toCalculation of Box Dimension of Fractal Functions Wiley 3rdedition 2007
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 7
1021
08172
06132
04092
02051
0001096
minus02029
minus0407
minus0611
minus0815
minus1019
minus1025
minus08177
minus06109
minus0404
minus01972
0009691
02165
04234
06303
08371
1044
Figure 6 Approximation of vector-function g(119905) = (cos(15119905)sin(119905)) by fractal interpolation function g⋆
2
15
1
05
0
minus05
minus1
minus15
minus2
minus1 minus00 minus0e minus04 minus32 0 02 04 06 30 1
Figure 7 Weierstrass function (blue one) and approximatingvector-function (red one)
sequences by fractal interpolation vector-functions whichare affine transformations withmatrix parameters Parameterfitting was a crucial part of approximation process We havefound appropriate parameter values of fractal interpolationvector-functions and illustrate it with several examples ofdifferent types of discrete data
We assume that fractal approximation is highly promisingcomputational tool for different types of data and it canbe used in many ways even in interdisciplinary fieldswith a quite high precision that allows us to apply fractalapproximation methods to a wide variety of curves smoothand nonsmooth alike
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The work was performed according to the Russian Gov-ernment Program of Competitive Growth of Kazan Federal
University The authors greatly acknowledge the anonymousreviewer for carefully reading the paper and providing con-structive comments that have led to an improved paper
References
[1] M F Barnsley ldquoFractal functions and interpolationrdquo Construc-tive Approximation vol 2 no 4 pp 303ndash329 1986
[2] M F Barnsley J Elton D Hardin and P Massopust ldquoHiddenvariable fractal interpolation functionsrdquo SIAM Journal onMath-ematical Analysis vol 20 no 5 pp 1218ndash1242 1989
[3] M F Barnsley and A N Harrington ldquoThe calculus of fractalinterpolation functionsrdquo Journal of Approximation Theory vol57 no 1 pp 14ndash34 1989
[4] P Massopust Interpolation and Approximation with Splines andFractals Oxford University Press Oxford UK 2010
[5] K Igudesman and G Shabernev ldquoNovel method of fractalapproximationrdquo Lobachevskii Journal of Mathematics vol 34no 2 pp 125ndash132 2013
[6] J E Hutchinson ldquoFractals and self-similarityrdquo Indiana Univer-sity Mathematics Journal vol 30 no 5 pp 713ndash747 1981
[7] M F Barnsley and L P Hurd Fractal Image Compression A KPeters Wellesley Mass USA 1993
[8] M Barnsley Fractals Everywhere Academic Press BostonMass USA 1988
[9] J R Magnus and H Neudecker ldquoMatrix differential calculuswith applications in statistics and econometricsrdquoApplied Math-ematical Sciences vol 8 no 144 pp 7175ndash7181 2014
[10] K Igudesman R Lavrenov and V Klassen Novel Approach toCalculation of Box Dimension of Fractal Functions Wiley 3rdedition 2007
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
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Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of