New Approach to Fractal Approximation of Vector-Functions

and Applied Analysis 3


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][2][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  2 by fractal approximation vector-functions.We find best values of matrix parameters by means of matrix differential calculus.Section 4 illustrates some examples.

Fractal Interpolation Vector-Functions
be the interpolation points.For all  = 1, , consider affine transformation Henceforth, small bold letters denote columns (rows) of length  and big bold letters denote matrices of  × .
Require that for all  the following conditions hold true: where matrices {D  }  =1 are considered as parameters.
Also notice that for all  operator   takes straight segment between ( 0 , x 0 ) and (  , x  ) to straight segment which connects points of interpolation ( −1 , x −1 ) and (  , x  ).
Let K be a space of nonempty compact subsets of R +1 , with Hausdorff metric.Define the Hutchinson operator [6] Φ : By the condition (2) Hutchinson operator Φ takes a graph of any continuous vector-function on segment [, ] to a graph of a continuous vector-function on the same segment.Thus, Φ can be treated as operator on the space of continuous vector-functions ([, ])  .For all  = 1, , denote In (1), substitute x to vector-function g().We have that Φ acts on ([, ])  according to Suppose that we consider all matrices D  as linear operators on R  .Furthermore, they are contractive mappings; that is, constant  ∈ [0, 1) exists such that for all k, w ∈ R  and  = 1,  we have Then, from (7) Function g ⋆ is called fractal interpolation vector-function.

Approximation
Henceforth, we assume that for all  = 1,  linear operator D  is contractive mapping with contraction coefficient  ∈ [0, 1).We approximate vector-function g ∈ ([, ])  by fractal interpolation vector-function g ⋆ constructed on points of interpolation {(  , x  )}  =0 .Thus, we need to fit matrix parameters D  to minimize the distance between g and g ⋆ .
We use methods that have been developed for fractal image compression [7].Denote Banach space of square integrated vector-functions on segment as Then from ( 7) and ( 8) and Remark 1 it follows that for all Abstract and Applied Analysis Thus, Φ :   2 [, ] →   2 [, ] is a contractive operator and g ⋆ is its fixed point.
Instead of minimizing ‖g − g ⋆ ‖ 2 we minimize ‖g − Φg‖ 2 that makes the problem of optimization much easier.The collage theorem provides validity of such approach [8].for all  ∈ .
Considering ( 4) and ( 6), rewrite (7) where Thus, we minimize the functional Lemma 4. Let f, h ∈   2 [, ] be square integrated vectorfunctions.Suppose that matrix ∫   hh  dt is nondegenerated.Matrix integration is implied to be componentwise.Then, the functional Proof.To prove it, we use matrix differential calculus [9].Consider Necessary condition of existence of functional Ψ extremum is dΨ(X, U) = 0 for all U ∈ R × .Since there is -linearity of functional dΨ(X, U), it is sufficient to prove dΨ(X, U) = 0 only for matrices U that consist of  2 −1 zeros and one unity.Therefore, we have  2 expressions for finding coefficients of matrix X.In matrix form these expressions are as follows: from which Hence, and then functional Ψ is convex one.Thus, the value X is absolute minimum of Ψ.

⋅ (g ∘ 𝑤
It is sufficient to apply (1) for constructing fractal interpolation vector-function after we find D  .
Consider several examples of approximation of discrete data.
In this case affine transformations (1) have the following form:   This example is taken from [10], where fractal approximation is used for approximate calculation of box dimension of fractal curves.

Conclusion
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.

Remark 7 .Example 8 .Example 9 .
Vectors c  in matrices of affine transformations (1) equal 0 (like in previous example).It means that fractal interpolation vector-function can be treated as attractor of classical affine IFS in R  .Next example is devoted to a circle g() = (cos , sin ),  ∈ [0, 2].Figure4shows the results.Here we also have two pictures; the first one illustrates initial vectorfunction and its approximation with 3 points and the second one with 5 points.In this case affine transformations (1) have the following form: Spiral of Archimedes g() = ( cos ,  sin ),  ∈ [0, 5], where the scheme is equal to the examples above, but here we use far more points of interpolation, as illustrated in Figure5.