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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.

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 [

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 [

This paper is multidimensional generalization of [

Let

Require that for all

Notice that

Also notice that for all

Let

For all

Suppose that we consider all matrices

Figure

Fractal interpolation vector-function

Henceforth, we assume that for all

We use methods that have been developed for fractal image compression [

Instead of minimizing

Let

Considering (

Let

To prove it, we use matrix differential calculus [

Consider

Hence,

From Lemma

Let us approximate vector-function

Calculate

Apply affine transformations from (

Vector-function

In this section, we approximate discrete data

It is necessary to use results of previous section. Approximate

Consider several examples of approximation of discrete data.

Let us approximate vector-function

In this case affine transformations (

Approximation of vector-function

Vectors

Next example is devoted to a circle

In this case affine transformations (

Approximation of vector-function

Spiral of Archimedes

Approximation of vector-function

Figure

Approximation of vector-function

The example illustrates approximation of graph of Weierstrass function

This example is taken from [

Weierstrass function (blue one) and approximating vector-function (red one).

In this paper, we have introduced new effective method of approximation of continuous vector-functions and vector sequences by fractal interpolation vector-functions, which are affine transformations with matrix parameters. Parameter fitting was a crucial part of approximation process. We have found appropriate parameter values of fractal interpolation vector-functions and illustrate it with several examples of different types of discrete data.

We assume that fractal approximation is highly promising computational tool for different types of data and it can be used in many ways, even in interdisciplinary fields, with a quite high precision that allows us to apply fractal approximation methods to a wide variety of curves, smooth and nonsmooth alike.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The work was performed according to the Russian Government Program of Competitive Growth of Kazan Federal University. The authors greatly acknowledge the anonymous reviewer for carefully reading the paper and providing constructive comments that have led to an improved paper.