Graph Directed Coalescence Hidden Variable Fractal Interpolation Functions

Fractal interpolation function (FIF) is a special type of continuous function which interpolates certain data set and the attractor of the Iterated function system (IFS) corresponding to the data set is the graph of the FIF. Coalescence Hidden-variable Fractal Interpolation Function (CHFIF) is both self-affine and non self-affine in nature depending on the free variables and constrained free variables for a generalized IFS. In this article graph directed iterated function system for a finite number of generalized data sets is considered and it is shown that the projections of the attractors on $\mathbb{R}^{2}$ is the graph of the CHFIFs interpolating the corresponding data sets.


Introduction
The concept of fractal interpolation function (FIF) based on an iterated function system (IFS) as a fixed point of Hutchinson's operator is introduced by Barnsley [?, ?].The attractor of the IFS is the graph of the fractal function interpolating certain data set.These FIFs are generally self-affine in nature.The idea has been extended to a generalized data set in R 3 such that the projection of the graph of the corresponding FIF onto R 2 provides a non self-affine interpolation function namely Hidden variable FIFs for a given data set {(x n , y n ) : n = 0, 1, . . ., N} [?].Chand and Kapoor [?], introduced the concept of Coalescence hidden variable FIFs which are both self-affine and non self-affine for generalized IFS.The extra degree of freedom is useful to adjust the shape and fractal dimension of the interpolation functions.In [?], Barnsley et al. proved existence of a differentiable FIF.The continuous but nowhere differentiable fractal function namely α-fractal interpolation function f α is introduced by Navascues as perturbation of a continuous function f on a compact interval I of R [?,?].Interested reader can see for the theory and application of α-fractal interpolation function f α which has been extensively explored by Navascues [?, ?, ?].
In [?] Deniz et al. considered graph-directed iterated function system for finite number of data sets and proved the existence of fractal functions interpolating corresponding data sets with graphs as the attractor of the GDIFS.
In the present work, generalized GDIFS for generalized interpolation data sets in R 3 has taken.It is shown that, corresponding to the data sets there exists CHFIFs whose graph is the projection on R 2 of the attractors of the GDIFS.

Preliminaries
2.1.Iterated Function System.Let X ⊂ R n and (X , d X ) be a complete metric space.Also assume H(X ) = {S ⊂ X ; S = Φ, S is compact in X } with the Hausdorff metric d H (A, B) defined as d H (A, B) = max{d X (A, B), d X (B, A)}, where d X (A, B) = max x∈A min y∈B d X (x, y) for any two sets A, B in H(X ).(H, d H ) is a complete metric space whenever (X , d X ) is complete.Let for i = 1, 2, . . ., N, w i : X → X are continuous maps then {X ; w i : i = 1, 2, . . ., N} is called an iterated function system (IFS).If the maps w i 's are contraction then, the set valued Hutchinson operator W : H(X ) → H(X ) defined by W (B) = N i=1 w i (B), where w i (B) := {w i (b) : b ∈ B} is also contraction.Then by Banach fixed point theorem, there exists a unique set A ∈ H(X ) such that A = W (A) = N i=1 w i (A).The set A is called the attractor associated with the IFS {X ; w i : i = 1, 2, . . ., N}.

2.2.
Fractal Interpolation Function.Let a set of interpolation points {(x i , y i ) ∈ I ×R : i = 0, 1, . . ., N} be given, where ∆ : for some 0 ≤ d < 1.Furthermore, let H i : K → R, i = 1, 2, . . ., N, be given continuous functions such that (3) for all x ∈ I and for all ξ 1 and ξ 2 in [g 1 , g 2 ], for some constitutes an IFS.Barnsley [?] proved that the IFS {K; W i : i = 1, 2, . . ., N} defined above has a unique attractor G where G is the graph of a continuous function f : I → R which obeys f (x i ) = y i for i = 0, 1, . .., N.This function f is called a fractal interpolation function (FIF) or simply fractal function and it is the unique function satisfying the following fixed point equation ( 6) The widely studied FIFs so far are defined by the iterated mappings where the real constants a i and d i are determined by the condition (1) as ( 8) and q i (x)'s are suitable continuous functions such that the conditions (3) and (4) hold.For each i, α i is a free parameter with |α i | < 1 and is called a vertical scaling factor of the transformation W i .Then the vector α = (α 1 , α 2 , . . ., α N ) is called the scale vector of the IFS.If q i (x) is taken as linear then the corresponding FIF is known as affine FIF (AFIF).

Coalescence FIF.
To construct a Coalescence Hidden-variable Fractal Interpolation Functions, a set of real parameters z i for i = 1, 2, . . ., N are introduced and the generalized interpolation data {(x i , y i , z i ) ∈ R 3 : i = 0, 1, . . ., N} is considered.Then define the maps w i : where, L i : I → I i , i = 1, 2, . . ., N are given in (7), and the functions Here α i , γ i are free variables with |α i | < 1, |γ i | < 1 and β i are constrained variable such that |β i | + |γ i | < 1.Then the generalized IFS {I × R 2 ; w i (x, y, z) : i = 1, 2, . . ., N} . The attractor G is the graph of a vector valued function f : 2.4.Graph-directed Iterated Function Systems.Let G = (V, E) be a directed graph where V denote the set of vertices and E is the set of edges.For all u, v ∈ V , let E uv denote the set of edges from u to v with elements e uv i , i = 1, 2, . . ., K uv where K uv denotes the number of elements of E uv .An iterated function system realizing the graph G is given by a collection of metric spaces (X v , ρ v ), v ∈ V , and of contraction mappings w uv i : X v → X u corresponding to the edge e uv i in the opposite direction of e uv i .An attractor (or invariant list) for such an iterated function system is a list of nonempty compact sets A u ⊂ X u such that for all u ∈ V , Then (X u ; w uv i ) is the graph directed iterated function system (GDIFS) realizing the graph G [?, ?].

Graph Directed Coalescence FIF
In this section, for a finite number of data sets, generalized graph-directed iterated function system (GDIFS) is defined so that projection of each attractor on R 2 is the graph of a CHFIF which interpolates the corresponding data set and call it as graph-directed coalescence hidden-variable fractal interpolation function.For simplicity, only two sets of data are considered.Let the two data sets as for all i = 1, 2, . . ., N and j = 1, 2, . . ., M. By introducing two set of real parameters z 1 i , z 2 j for i = 1, 2, . . ., N and j = 1, 2, . . ., M, consider the two generalized data set as give a picture.To construct a generalized GDIFS associated with the data D r , (r = 1, 2) and realizing the graph G consider the functions w rs n : R 3 → R 3 defined as From each of the above conditions, the following can derive respectively. (10) From the linear system of equations ( 10), ( 11), ( 12) and (13) the constants a rs i , b rs i , c rs i , d rs i , e rs i and f rs i for r, s ∈ {1, 2}, i = 1, 2, . . ., K rs are determined as follows The following theorem shows that each maps w rs n is contraction with respect to metric equivalent to the Euclidean metric and ensures the existence of attractors of generalized GDIFS.
Theorem 3.1.Let {R 3 ; w rs n , n = 1, 2, . . ., K rs } be the generalized GDIFS defined above realizing the graph and associated with the data sets D r , (r = 1, 2) which satisfy (9).If |α rs n | < 1, |γ rs n | < 1 and β rs n is chosen such that |β rs n | + |γ rs n | < 1 for all r, s ∈ {1, 2} and n = 1, 2, . . ., K rs .Then there exists a metric δ on R 3 equivalent to the Euclidean metric, such that the GDIFS is hyperbolic with respect to δ.In particular, there exists non empty compact sets G r such that Proof.Proof follows in the similar line of Theorem 2.1.1,[?] and using above condition (9).
Following is the main result regarding existence of coalescence Hidden-variable FIFs for generalized GDIFS.
Theorem 3.2.Let G r , r ∈ V be the attractors of the generalized GDIFS as in Theorem 3.1.Then G r , r ∈ V is the graph of a vector valued continuous function f r : Proof.Consider the vector valued function spaces respectively, where .denotes a norm on R 2 .Since (F , d F ) and (H, d H ) are complete metric spaces, then (F × H, d) is also a complete metric space where Following are the affine maps.