Abstract and Applied Analysis

Volume 2015, Article ID 278313, 7 pages

http://dx.doi.org/10.1155/2015/278313

## New Approach to Fractal Approximation of Vector-Functions

^{1}Geometry Department, Lobachevskii Institute of Mathematics and Mechanics, Kazan (Volga Region) Federal University, Kazan 420008, Russia^{2}Kazan (Volga Region) Federal University, IT-Lyceum of Kazan University, Kazan 420008, Russia^{3}Department of Autonomous Robotic Systems, High School of Information Technologies and Information Systems, Kazan (Volga Region) Federal University, Kazan 420008, Russia

Received 19 August 2014; Revised 24 January 2015; Accepted 8 February 2015

Academic Editor: Poom Kumam

Copyright © 2015 Konstantin Igudesman et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper introduces new approach to approximation of continuous vector-functions and vector sequences by fractal interpolation vector-functions which are multidimensional generalization of fractal interpolation functions. Best values of fractal interpolation vector-functions parameters are found. We give schemes of approximation of some sets of data and consider examples of approximation of smooth curves with different conditions.

#### 1. Introduction

It is well known that interpolation and approximation are an important tool for interpretation of some complicated data. But there are multitudes of interpolation methods using several families of functions: polynomial, exponential, rational, trigonometric, and splines to name a few. Still it should be noted that all these conventional nonrecursive methods produce interpolants that are differentiable a number of times except possibly at a finite set of points. But, in many situations, we deal with irregular forms, which can not be approximate with desired precision. Fractal approximation became a suitable tool for that purpose. This tool was developed and studied in [1–3].

We know that such curves as coastlines, price graphs, encephalograms, and many others are fractals since their Hausdorff-Besicovitch dimension is greater than unity. To approximate them, we use fractal interpolation curves [1] and their generalizations [4] instead of canonical smooth functions (polynomials and splines).

This paper is multidimensional generalization of [5]. In Section 2, we consider fractal interpolation vector-functions which depend on several matrices of parameters. Example of such functions is given. In Section 3, we set the optimization problem for approximation of vector-function from by fractal approximation vector-functions. We find best values of matrix parameters by means of matrix differential calculus. Section 4 illustrates some examples.

#### 2. Fractal Interpolation Vector-Functions

Let be a nonempty interval; let and be the interpolation points. For all , consider affine transformationHenceforth, small bold letters denote columns (rows) of length and big bold letters denote matrices of .

Require that for all the following conditions hold true:Then, Solving the system, we havewhere matrices are considered as parameters.

*Remark 1. *Notice that .

Also notice that for all operator takes straight segment between and to straight segment which connects points of interpolation and .

Let be a space of nonempty compact subsets of , with Hausdorff metric. Define the Hutchinson operator [6] By the condition (2) Hutchinson operator takes a graph of any continuous vector-function on segment to a graph of a continuous vector-function on the same segment. Thus, can be treated as operator on the space of continuous vector-functions .

For all , denoteIn (1), substitute to vector-function . We have that acts on according to

Suppose that we consider all matrices as linear operators on . Furthermore, they are contractive mappings; that is, constant exists such that for all and we haveThen, from (7), it follows that operator is contraction with contraction coefficient on Banach space , where . By the fixed-point theorem, there exists unique vector-function such that and for all we have Function is called fractal interpolation vector-function. It is easy to notice that if , , and , then passes through points of interpolation. In this case functions are called prefractal interpolation vector-functions of order .

*Example 2. *Figure 1 shows fractal interpolation vector-function of plane. Here , , and and , , and . Values of matrices and are