Fractal Interpolation Representation of A Class of Quadratic Functions

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1 ISSN (print, (online International Journal of Nonlinear Science Vol.24(2017 No.3, pp Fractal Interpolation Representation of A Class of Quadratic Functions Chengbin Shen, Zhigang Feng Facult of Science, Jiangsu Universit, Zhenjiang, Jiangsu, P.R. China (Received 2 Januar 2017, accepted 15 Ma 2017 Abstract: Fractal interpolation functions are important research contents in fractal theor. These functions generated b certain iterative function sstems often have complex graphics and non-integer dimension. In general, it is difficult for us to get analtic expressions of fractal interpolation functions. In this paper, we investigate the representation of a class of special fractal interpolation functions. From the iterative process, we find that this fractal interpolation function is essentiall a parabola when vertical scale factor is a given constant. Furthermore, we find that a class of quadratic functions can be represented as certain fractal interpolation functions, which paved the wa for the further research of the analtic expressions of the fractal interpolation functions. Kewords: Quadratic Function; self affine; fractal interpolation function. 1 Introduction Before the emergence of fractal theor, we use straight lines, boxes, rounds and other tools to describe the usual objects. But for man extremel irregular and disorderl objects, we can do nothing. The American mathematician Barnsle [1] proposed the fractal interpolation function in The image of the function can approximatel describe the object that the Euclidean geometr is not well described [2], such as the coast forest top fluctuant curve, the contours of the mountain, the cloud shape and so on. As long as the vertical scaling factor is required to transform, it can generate an curve whose dimension is between 1 and 2. Lebesgue [3] have given important ideas related to dimensions. On the basis of the measure, Hausdorff [4] introduced the Hausdorff dimension in 1919, which enhances the practicalit. But Hausdorff dimension is difficult to calculate, and it has little actual background. This brought great obstacles to our follow-up stud. Then Bouligand raised the box dimension in the last centur. It can be calculated b experimental approximation, so its practical use more widel. As a new fitting method, the fractal interpolation has caused wide attention of applied mathematicians. In the numerical calculation, computational geometr, computer graphics and other fields, it has a ver wide range of applications. In the theor of functions, interpolation function approximation is one of the main contents. In Ref. [5], the construction of fractal interpolation function is introduced. In man cases, the fractal interpolation function (FIF [6] is often generated b iterated function sstem (IFS. The smoothness and continuit of the fractal interpolation function have been discussed adequatel in [7-9]. Li [10] depicted the smoothness of FIF and gave the relationship between the various parameters of IFS and the smoothness of FIF. Barnsle first gave the formula for calculating the box dimension, and enhanced the practical significance of fractal geometr. On this basis, Sha and Ruan [6] rejected the specific conditions, which further strengthening the application value. After introducing the concept of variabilit, the dimension formula of the continuous function image on the plane can be given directl b the variance estimation. Recentl, the methods of image generation, the calculation of dimensions, and graphics simulation, have been the ke research direction of the fractal theor and practical application[11-13]. However, few scholars have paid attention to the representation of fractal interpolation, because of its difficult and complexit. It is such functions without obvious expressions, so that our research process is relativel slow. This paper attempts to give an expression to a special case of FIF, and extends to the general case. Corresponding author. address: @qq.com Copright c World Academic Press, World Academic Union IJNS /987

2 176 International Journal of Nonlinear Science, Vol.24(2017, No.3, pp Figure 1: The stable region of d 1 d 2 2/3. Figure 2: The stable region of d 1 d 2 1/4. 2 Fractal Interpolation function Let a set of interpolation nodes{(x i, i R 2, i 0, 1, 2,, N},N 2 be given and 0 x 0 < x 1 < < x N 1. Define mapping ω i : [0, 1] R [0, 1] R, ω i ( x ( Li (x F i (x,, i 0, 1, 2,, N, (2.1 L i (x satisfies L i (x 0 x (i 1, L i (x N x i. (2.2 Let 1 < d i < 1, F i (x, : [x 0, x N ] R R satisf x [x 0, x N ], 1, 2 R and then F i (x, 1 F i (x, 2 d i 1 2, F i (x 0, 0 (i 1, F i (x N, N i, (i 1, 2,, N. (2.3 B the formula (2.1, the function f : [x 0, x N ] R can be uniquel determined, and satisfies When L i (x, F i (x, are linear functions, ω i has the following form: ω i ( x f(l i (x F i (x, f(x, i 1, 2,, N. (2.4 ( ai 0 c i d i ( x ( ei +, i 1, 2,, N. (2.5 f i B the formulas (2.2 and (2.3, we can calculate the coefficient of iterated function sstem, parameters d i is vertical scale factor a i (x i x (i 1 /(x N x 0, e i (x N x (i 1 x 0 x 1 /(x N x 0, c i ( i (i 1 /(x N x 0 d i ( N 0 /(x N x 0, f i (x N (i 1 x 0 i /(x N x 0 d i (x N 0 x 0 N /(x N x 0. There is a continuous function whose image: G {(x, f(x x [x 0, x N ]} is the invariant set of iterated function sstem ω i [6]. The selection of the vertical compression factor has a considerable influence on the shape of the image, as shown in F ig.1. Ruan et.al [14] have proved that when N 2, x 0 0, x 1 1 2, x 2 1, 0 2 0, 1 1, d 1 d 2 1 4, the FIF obtained b Iterated function sstem (2.5 is f(x 4x 2 + 4x, x [0, 1], as shown in Fig.2. (2.6 IJNS for contribution: editor@nonlinearscience.org.uk

3 C. Shen and Z. Feng: Fractal Interpolation Representation of A Class of Quadratic Functions Main Results Theorem 1 [15] : Let x 0 < x 1 < < x N be given, then L i (x is an affine mapping and satisfies (2.2, Let If a i (x i x i 1 /(x N x 0, F i (x, d i + q i (x, i 1, 2,, N. F i,k (x, d i + qi k(x a k, i d i < a n i, q i C n [x 0, x N ], i 1, 2,, N, q (0 i q i ; 0,k qk 1 (x 0 a k 1 d, N,k qk N (x N 1 a k N d, k 1, 2,, n. N F i 1,k (x N, N,k F i,k (x 0, 0,k, then F C n [x 0, x N ] determined b {L i (x, F i,k (x, } N i1 and f k determined b {L i (x, F i,k (x, } N i1 are both fractal interpolation functions. Theorem 2 When N 3, let x 0 0, x 1 1 3, x 2 2 3, x 3 1, 0 3 0, 1 2 1, d 1 d 2 d 3 1 9, the FIF obtained b IFS (2.5 is f(x 9 2 x x, x [0, 1]. Proof. According to the conditions of the theorem, the concrete form of IFS can be obtained b formula (2.6: ( ( ( ( x 1/3 0 x x/3 ω 1, 1 1/9 x + /9 ( ( ( ( ( x 1/3 0 x 1/3 1/3x/3 ω 2 +, 0 1/ /9 ( ( ( ( ( x 1/3 0 x 2/3 2/3x/3 ω /9 1 1 x + /9 (1 Let L 1 (x x/3, L 2 (x x/3 + 1/3, L 3 (x x/3 + 2/3, F 1 (x, /9 + q 1 (x, F 2 (x, /9 + q 2 (x, F 3 (x, /9 + q 3 (x, q 1 (x x, q 2 (x 1, q 3 (x x 1. On account of L i (x, F i (x, satisfing (2.2and(2.3, f : [x 0, x N ] R based on {L i (x, F i (x, } 3 i1 satisfies B calculation, we can get f(l i (x F i (x, f(x, i 1, 2, 3, F i,1 (x, d i + q i (x a i, i 1, 2, 3, 0,1 q 1(x 0 a 1 d 1, 3,1 q 3(x 3 a 3 d 3. F 1,1 (x, 3 + 3, F 2,1(x, 3, F 3,1(x, 3 3, 0,1 9 2, 3,1 9 2, and F 1,1 (x 3, 3,1 F 2,1 (x 0, 0,1 3 2, F 2,1(x 3, 3,1 F 3,1 (x 0, 0, Therefore according to Theorem 1, a fractal interpolation function which satisfies h(l i (x F i,1 (x, h(x, i 1, 2, 3, f h (3.2 IJNS homepage:

4 178 International Journal of Nonlinear Science, Vol.24(2017, No.3, pp can be determined b {L i (x, F i (x, } 3 i1. Then r(x 9x + 9/2 is a straight line equation, passing through two points: (x 0, 0,1 (0, 9/2, (x 3, 3,1 (1, 9/2. Due to thus the two points 1,1 3/2, 2,1 3/2, (x 1, 1,1 (1/3, 3/2, (x 2, 2,1 (1/3, 3/2 are on line r(x 9x + 9/2. That is, i,1 r(x i, i 1, 2, 3. For x [x 0, x 3 ], there are λ 1 and λ 2 satisfing Therefore, we can get the following results: λ 1 + λ 2 1, λ 1 0, λ 2 0, x λ 1 x 0 + λ 2 x 3, i 1, 2, 3. r(l i (x r(l i (λ 1 x 0 + λ 2 x 3 r(λ 1 x i 1 + λ 2 x i λ 1 r(x i 1 + λ 2 r(x i, F i,1 (x, r(x F i,1 (x, r(λ 1 x 0 + λ 2 x 3 F i,1 (x, λ 1 r(x 0 + λ 2 r(x 3 λ 1 F i,1 (x 0, r(x 0 + λ 2 F i,1 (x 3, r(x 3 λ 1 i 1,1 + λ 2 i,1 λ 1 r(x i 1 + λ 2 r(x i. Thus : r(l i (x F i,1 (x, r(x, x [x 0, x 3 ], i 1, 2, 3. According to formula (3.2, we can get Thus f (x h(x r(x 9x + 9/2, f(0 0. This ends the proof. f(x 9 2 x2 + 9 x, x [0, 1]. 2 Theorem 3 About f(x Ax 2 + Ax, x [0, 1](A > 0, it alwas exists a definition mapping {L i (x, F i (x, } N i1, N 2,and a set of interpolation nodes that can determine an iterated function sstem whose attractor G is the graph of this quadratic function. Proof. Let x i i N, i A(iN i2 N, i 1, 2, 3,, N and give interpolation points: 2 It is eas to prove that: (x i, i R 2, i 1, 2, 3,, N, let d 1 d 2 d N 1 N 2. L i (x 1 N x + i 1 N, F i(x, 1 N 2 + q i(x, A(N 2i + 1 [(i 1]N (i 12 q i (x N 2 + A N 2, i 1, 2, 3,, N. The following process is similar to Theorem 2. IJNS for contribution: editor@nonlinearscience.org.uk

5 C. Shen and Z. Feng: Fractal Interpolation Representation of A Class of Quadratic Functions Conclusions In this paper, we investigate a class of special fractal interpolation functions obtained b the unique attractor of iterated function sstems. From the iterative process, we find that this fractal interpolation function is essentiall a parabola when vertical scale factor d1/4. Then we find that all of the quadratic functions like f(x Ax 2 + Ax, x [0, 1], A > 0 can be represented b certain fractal interpolation functions which are separated b an isometr. Fractal interpolation functions can be used to get the basic line and the quadratic function, and provide a reference for the further stud of other functions (such as polnomials, etc. represented b fractal interpolations. Ackowledgments This paper is supported b the National Natural Science Foundation of China (No: , and the Major Project of Natural Science Foundation of Jiangsu Province Colleges and Universities (14KJA References [1] M. F. Barnsle. Fractal functions and interpolation. Constructive Approximation, 2(4(1986: [2] B. Mandelbrot. The fractal geometr of nature. New York: W. H. Freeman [3] H. Lebesgue. Sur la non-applicabilité de deux domaines appartenant respectivement à des espaces àn etn+p dimensions. Mathematische Annalen, 70(2(1991: [4] F. Hausdorff. Dimension und äußeres Maß. Mathematische Annalen. 79(1(1918: [5] M. F. Barnsle, J. H. Elton and D. P. Hardin. Recurrent interated function sstems. Constructive Approximation, 5(1989: [6] Z. Sha and H. J. Ruan. Fractal and fitting. Hangzhou: Zhejiang Universit Press [7] M. F. Barnsle, J. H. Elton and D. P. Hardin. Hidden variable fractal interpolation functions. Siam Journal on Mathematical Analsis, 20(5(1989: [8] H. Y. Wang. On smoothness for a class of fractal interpolation surface. Fractals, 14(3(2006: [9] H. Y. Wang. Sensitivit analsis for hidden variable fractal interpolation functions and their moments. Fractals, 17(2(2009: [10] H. D. Li, Z. L. Ye and X. S. Gao. On the continuit and differentiabilit of fractal interpolation function. Applied Mathematics and Mechanics,23(4(2002. [11] P. Bouboulis, L. Dalla and V. Drakopoulos. Construction of recurrent bivariate fractal interpolation surfaces and computation of their box-counting dimension. Journal of Approximation Theor, 141(2006: [12] H. Y. Wang, S. Z. Yang and X. J. Li. Error analsis for bivariate fractal interpolation functions generated b 3D perturbed iterated function sstems. Computers Mathematics with Applications, 56(2008: [13] Z. G. Feng and X. Q. Sun. Box-counting Dimensions of Fractal Interpolation Surfaces Derived from Fractal Interpolation Functions. J. Math. Anal. Appl., 412(1(2014: [14] H. J. Ruan, Z Sha and W. Y. Su. Parameter identification problem of the fractal interpolation functions. Numerical Mathematics A Journal of Chinese Universities English Series, 12(2(2003: [15] M. F. Barnsle and A. N. Harrington. The calculus of fractal interpolation functions. Journal of Approximation Theor, 57(1(1989: IJNS homepage:

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