Research Article New Approach to Fractal Approximation of …downloads.hindawi.com/journals/aaa/2015/278313.pdf · 2019-07-31 · Research Article New Approach to Fractal Approximation

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)


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


119870= 119887 by fractal

interpolation vector-function g⋆ constructed on points ofinterpolation 119883 = (119905

119894 x119894)119873

119894=0 119886 = 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))


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


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


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


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


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


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)


10

05

minus05

minus10

02 04 06 08 10


Thus Φ 119871119872

2[119886 119887] rarr 119871

119872




2




119889 (119909 119909⋆) le

119889 (119909 119879 (119909))

1 minus 119888(13)

for all 119909 isin 119883


(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) =


) 119905 + (119905119899x119899minus1


x119899)

119905119899minus 119905119899minus1

k119899(119905) =

(x119873minus x0) 119905 + (119905

119899x0minus 119905119899minus1

x119873)

119905119899minus 119905119899minus1

119908119899(119905) =


119887)

119905119899minus 119905119899minus1

(15)


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)






Ψ R119872times119872

997888rarr R Ψ (X) = int119887

119886





dΨ (XU) = d(int119887

119886


= d(int119887

119886


= int

119887

119886


= 2int

119887

119886

(minush119879U119879f + h119879U119879Xh) d119905(18)



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)



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)


1

09

08

07

06

05

04

03

02

01

7644e minus 31

minus1

minus08

minus06

minus04

minus02

02

04

06

08 1

2165eminus15






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)


D1= (

1

40

minus3

8

1

8

) D2= (

1

40

3

8

1

8

) (24)


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)




(119911119898w119898)119870

119896=0 119886 = 119911


119870= 119887 by fractal


119894 x119894)119873

119894=0 119886 = 119905


119873= 119887



sum

119896=0

1003816100381610038161003816w119896 minus g⋆ (119911119896)1003816100381610038161003816

2

(26)



119896w119896) isin 119885 119911

119896


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))


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)




sdot (g ∘ 119908119899(119911119896) minus k119899(119911119896))119879

)

minus1

119899 = 1119873

(27)


119899



2) (119905 minus 1)2(119905 + 1)




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)






1198601(

119905

1199091

1199092

) = (

05 0 0

minus03180 00006 02128


)(

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)



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)


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)




infin

119899=0(12)119899 cos(21205874119899119909)



5 Conclusion



1021

08172

06132

04092

02051

0001096

minus02029

minus0407

minus0611

minus0815

minus1019

minus1025

minus08177

minus06109

minus0404

minus01972

0009691

02165

04234

06303

08371

1044


2

15

1

05

0

minus05

minus1

minus15

minus2







Acknowledgments



References


















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










10

05

minus05

minus10

02 04 06 08 10


Thus Φ 119871119872

2[119886 119887] rarr 119871

119872




2




119889 (119909 119909⋆) le

119889 (119909 119879 (119909))

1 minus 119888(13)

for all 119909 isin 119883


(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) =


) 119905 + (119905119899x119899minus1


x119899)

119905119899minus 119905119899minus1

k119899(119905) =

(x119873minus x0) 119905 + (119905

119899x0minus 119905119899minus1

x119873)

119905119899minus 119905119899minus1

119908119899(119905) =


119887)

119905119899minus 119905119899minus1

(15)


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)






Ψ R119872times119872

997888rarr R Ψ (X) = int119887

119886





dΨ (XU) = d(int119887

119886


= d(int119887

119886


= int

119887

119886


= 2int

119887

119886

(minush119879U119879f + h119879U119879Xh) d119905(18)



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)



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)


1

09

08

07

06

05

04

03

02

01

7644e minus 31

minus1

minus08

minus06

minus04

minus02

02

04

06

08 1

2165eminus15






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)


D1= (

1

40

minus3

8

1

8

) D2= (

1

40

3

8

1

8

) (24)


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)




(119911119898w119898)119870

119896=0 119886 = 119911


119870= 119887 by fractal


119894 x119894)119873

119894=0 119886 = 119905


119873= 119887



sum

119896=0

1003816100381610038161003816w119896 minus g⋆ (119911119896)1003816100381610038161003816

2

(26)



119896w119896) isin 119885 119911

119896


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))


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)




sdot (g ∘ 119908119899(119911119896) minus k119899(119911119896))119879

)

minus1

119899 = 1119873

(27)


119899



2) (119905 minus 1)2(119905 + 1)




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)






1198601(

119905

1199091

1199092

) = (

05 0 0

minus03180 00006 02128


)(

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)



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)


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)




infin

119899=0(12)119899 cos(21205874119899119909)



5 Conclusion



1021

08172

06132

04092

02051

0001096

minus02029

minus0407

minus0611

minus0815

minus1019

minus1025

minus08177

minus06109

minus0404

minus01972

0009691

02165

04234

06303

08371

1044


2

15

1

05

0

minus05

minus1

minus15

minus2







Acknowledgments



References


















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1

09

08

07

06

05

04

03

02

01

7644e minus 31

minus1

minus08

minus06

minus04

minus02

02

04

06

08 1

2165eminus15






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)


D1= (

1

40

minus3

8

1

8

) D2= (

1

40

3

8

1

8

) (24)


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)




(119911119898w119898)119870

119896=0 119886 = 119911


119870= 119887 by fractal


119894 x119894)119873

119894=0 119886 = 119905


119873= 119887



sum

119896=0

1003816100381610038161003816w119896 minus g⋆ (119911119896)1003816100381610038161003816

2

(26)



119896w119896) isin 119885 119911

119896


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))


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)




sdot (g ∘ 119908119899(119911119896) minus k119899(119911119896))119879

)

minus1

119899 = 1119873

(27)


119899



2) (119905 minus 1)2(119905 + 1)




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)






1198601(

119905

1199091

1199092

) = (

05 0 0

minus03180 00006 02128


)(

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)



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)


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)




infin

119899=0(12)119899 cos(21205874119899119909)



5 Conclusion



1021

08172

06132

04092

02051

0001096

minus02029

minus0407

minus0611

minus0815

minus1019

minus1025

minus08177

minus06109

minus0404

minus01972

0009691

02165

04234

06303

08371

1044


2

15

1

05

0

minus05

minus1

minus15

minus2







Acknowledgments



References


















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










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)




sdot (g ∘ 119908119899(119911119896) minus k119899(119911119896))119879

)

minus1

119899 = 1119873

(27)


119899



2) (119905 minus 1)2(119905 + 1)




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)






1198601(

119905

1199091

1199092

) = (

05 0 0

minus03180 00006 02128


)(

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)



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)


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)




infin

119899=0(12)119899 cos(21205874119899119909)



5 Conclusion



1021

08172

06132

04092

02051

0001096

minus02029

minus0407

minus0611

minus0815

minus1019

minus1025

minus08177

minus06109

minus0404

minus01972

0009691

02165

04234

06303

08371

1044


2

15

1

05

0

minus05

minus1

minus15

minus2







Acknowledgments



References


















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










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)


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)




infin

119899=0(12)119899 cos(21205874119899119909)



5 Conclusion



1021

08172

06132

04092

02051

0001096

minus02029

minus0407

minus0611

minus0815

minus1019

minus1025

minus08177

minus06109

minus0404

minus01972

0009691

02165

04234

06303

08371

1044


2

15

1

05

0

minus05

minus1

minus15

minus2







Acknowledgments



References


















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1021

08172

06132

04092

02051

0001096

minus02029

minus0407

minus0611

minus0815

minus1019

minus1025

minus08177

minus06109

minus0404

minus01972

0009691

02165

04234

06303

08371

1044


2

15

1

05

0

minus05

minus1

minus15

minus2







Acknowledgments



References


















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article New Approach to Fractal Approximation of …downloads.hindawi.com/journals/aaa/2015/278313.pdf · 2019-07-31 · Research Article New Approach to Fractal Approximation