A Method of the Best Approximation by Fractal Function

: We present a method constructing a function which is the best approximation for given data and satisfiesthe given self-similar condition. For this, we construct a space F of local self-similar fractal functions and show its properties. Next we present a computational scheme constructing the best fractal approximation in this space and estimate an error of the constructed fractal approximation. Our best fractal approximation is a fixed point of some fractal interpolation function


Introduction
Fractal approximation has been applied to model the objects which have fractal characteristics in nature. Fractal functions whose graphs are fractal sets have been widely used in approximation theory, signal processing, interpolation theory,computer graphics and so on. Hence, constructions of fractal functions and fractal approximation have been studied in many papers.
Constructions of fractal functions by fractal interpolation have been introduced by many researchers. A construction of one variable fractal interpolation functions by the iterated function system (IFS) with a data set on R was studied in [1,2,14], where the constructed fractal functions were self-similar ones. The construction was generalized in [3,4,17], which constructed local self-similar fractal functions. Constructions of bivariate fractal interpolation functions (BFIFs) have been studied in [5-11, 13, 17]. A construction of BFIFs by fractal interpolation on R was presented in [5,17] and self-affine fractal interpolation functions were constructed by IFS with a data set on a triangular domain in [12]. Constructions of self-similar BFIFs in [9,11] and self-affine BFIFs in [10,13] by IFS with a data set on a rectangular grid were introduced. In [6], local self-similarBFIFs were constructed by the recurrent iterated function system(RIFS) on a rectangular grid. A construction of local self-similar fractal interpolation functions in R n was studied in [4].
To construct fractal interpolation we need a data set } , , and a set of scale parameters The fractal property of the graph of the interpolation function is determinated by those data. Let a division of the interval and scale parameters be given, that is, a fractal property of the function be given. If the number of experimental data is more than the number of the interval division, then we can not construct the fractal interpolation for the data using fractal interpolation theory.
So we assume that a division of the interval and scale parameters be given (that is, a fractal property of the function) and study the problem constructing the best fractal approximation for the data set } , , , where n m > ( n is the number of the interval division).
In [15,18], constructionsofthebest approximation of functions by the fractal functions were presented, respectively. Butthe continuity of the approximation was not guaranteed then. The best fractal approximation of a continuous function in 2 L space was introduced in [16]. In [7], a space of differentiable fractal interpolation functions was constructed and it was proved that the constructed space is the reproducing kernel Hilbert space.
We construct a space of fractal interpolation functions with a given division of the interval and scale parameters and find a function satisfying some approximation condition for data }, , ,

A Space of Local Self-Similar Fractal Functions and a Space of Contractive Operators
In this section, we construct a space of local self-similar fractal functions and a space of contraction operators which are isomorphic to each other. Let ( where functions , and satisfy the following conditions: Define a space of functions satisfying the equations (4), (5) by F . The graph of F ∈ f has a local self-similarity and And for In fact, the existence and uniquenssof f areensured by the existence and uniquenss of the recurrent fractal interpolation function ( [3]).
This shows that the mapping 1 : is a bijection. We can easily check that the mapping Ψ is linear. Hence F and The space F is a Banach space with the norm ∞ ⋅ || || .
For a satisfies the following conditions:     T  T  T  T  T  T  T  i  i  i  i  i  i  i  T  T  T  T  T  T  i  i  i  i  i and we get Hence, T and 1 R n+ are isomorphic, which means that the dimension of T is n+1.
By the isomorphic relation, whose fixed point is 0 Theorem 1. Let F and T be the linear spaces constructed above. Then they are isomorphic.
Proof. This follows from Lemmas 1 and 2.
Denote the isomorphism of F to T by Ψ . Note that for

Construction of LSFA of a Data Set
In this section, we prove that there exists the least squares fractal approximation f in F of a data setand present an algorithm for finding f by calculating approximately the contraction operatorTin T corresponding to f.
Let P be a data set given by

Proof. Define an operator
We get a normal equation are fractal functions in (34), it needs enormous operations. Therefore, we consider an algorithm for calculating the approximation of The operator m T is given by (15) We find a * m T such that This problem is a minimization problem of a multi-variable function with unknown , is our best fractal approximation.

Estimation for Errors of the Approximation
In this section, we consider a relation between m T and T and estimatean error betweenthe approximation solution * T f and given data. Because m X is equivalent to and get a diagram that shows the relation between T and m T (see Fig 1).