# Uniform Continuity of Fractal Interpolation Function

Authors: Xuezai Pan; Minggang Wang; Xudong Shang
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

In order to research analysis properties of fractal interpolation function generated by the iterated function system defined by affine transformation, the continuity of fractal interpolation function is proved by the continuous definition of function and the uniform continuity of fractal interpolation function is proved by the definition of uniform continuity and compactness theorem of sequence of numbers or finite covering theorem in this paper. The result shows that the fractal interpolation function is uniformly continuous in a closed interval which is from the abscissa of the first interpolation point to that of the last one.


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## Body

## 1. Introduction

In 1960s, fractal geometry was regarded as a new interdiscipline firstly discovered by American mathematician Mandlebrot [1–5]. On the one hand, because most extremely irregular graphics in nature and very irregular social phenomenon are researched in the fractal geometry field, fractal geometry is called natural geometry. Therefore, fractal geometry is applied in almost all fields, such as mathematics, physics, chemistry, engineering, social science, and art [6–9]. On the other hand, for research on the natural properties of fractal itself, many methods used in researching fractals have been found by experts from 1960s to now, for example, fractal dimension method [10], multifractal spectrum method [11], adaptive fuzzy output-feedback method of nonlinear system [12], and nonlinear iterated method [13]. Especially, the fractal interpolation function method has been paid more and more attention by mathematicians. The theory of fractal interpolation function generated by the iterated function system defined by affine transformation was firstly proposed by Barnsley [14–16] and Massopust [17, 18]. They found that any part of a fractal graphic is similar to the whole, so they used mathematical language to express the similar iterated process. That is to say, first, the iterated function system consisting of affine transformation is defined and it is proved that the iterated function system has a unique attractor that is the fixed point. Second, according to the theory of iterated function system and self-similar theory, complex fractal graphic, the graphic of fractal interpolation curve (refer with Figure 1), or fractal interpolation surface (refer with Figure 2) can be generated by computer program. Finally, the dimension theory and integrability of fractal interpolation function have been studied by Barnsley and Massopust.Figure 1
Fractal interpolation curve.Figure 2
Fractal interpolation surface.Based on the research of fractal interpolation function above, the continuity and uniform continuity of the fractal interpolation function-generated iterated function system-defined affine mapping are proved in the paper.


## 2. Main Concepts and Lemmas

Definition 1.
(see [19, 20]). Let f be a function defined on interval I. If ∀x¯∈I and ∀ε>0, there is a δ>0, so that for any x∈I and x−x¯<δ⟹fx−fx¯<ε, we call that f is continuous on the point x¯ and the f is called continuous function on I.Definition 2.
(see [19, 20]). Let f be a function defined on interval I. If ∀ε>0 and ∃δ>0, so that for any x′,x″∈I, x′−x″<δ⟹fx′−fx″<ε, the f is called uniformly continuous function on I.Definition 3.
(see [19]). Let S be a point subset on real number set R. Let H be a open-interval set; in other words, each element in H is the open-interval form of α,β. H is called an open covering of S or H covers S, if any point in S is contained in at least an open interval in H. Furthermore, the number of open intervals in H is infinite (or finite) and we call that H is an infinite open covering of S (or a finite open covering of S).Definition 4.
(see [14–16]). A (hyperbolic) iterated function system consists of a complete metric space (X,d) together with a finite set of contraction mappings wi: X⟶X, with respective contractivity mapping factors si, for i=1,2,…,n. The abbreviation “IFS” is used for “iterated function system.” The notation for the IFS just announced is {X;wi,i=1,2,…,n} and contractivity factor is s = maxsi:i=1,2,…,n.Definition 5.
(see [14]). Let xi,yi∈R2;i=0,1,2,…,n be a set of points, where x0<x1<x2<⋯<xn. An interpolation function corresponding to this set of data is a function f: x0,xn⟶R such that(1)fxi=yi,i=1,2,…,n.
The pointsxi,yi are called the interpolation points. It is called that the function of f interpolates the data and that the graph of f passes through the interpolation points.Lemma 1 (see [19, 20]).
If a sequencexnn=1∞ is bounded, the sequence xn has a convergent subsequence xnkk=1∞.Lemma 2 (see [19, 20]).
Let H be a infinite open covering of closed intervala,b; then, there exist finite open intervals selected from H to cover a,b.Lemma 3 (see [14–18]).
Letn be a positive integer greater than 1. Let R2;wi,i=1,2,…,n denote the IFS defined above, associated with the data set(2)xi,yi∈R2;i=0,1,2,…,n.Let the vertical scaling factordi obey 0≤di<1 for i=1,2,…,n. Then, there is a metric d on R2, equivalent to the Euclidean metric, such that the IFS is hyperbolic with respect to d. In particular, there is a unique nonempty compact set G⊂R2, such that(3)G=⋃i=1nwiG.In particular, anIFS of the form R2;wi,i=1,2,…,n is considered, where the mapping is an affine transformation of the special structure(4)wixy=ai0cidixy+eifi,The transformations are constrained by the data according to(5)wix0y0=xi−1yi−1,wixnyn=xiyi,for,i=1,2,…,n,ai,ei,ci,andfi can be solved from equations (4)–(5) in terms of the data and vertical scaling factor di:(6)ai=xi−xi−1xn−x0,(7)ei=xnxi−1−x0xixn−x0,(8)ci=yi−yi−1xn−x0−diyn−y0xn−x0,(9)fi=xnyi−1−x0yixn−x0−dixny0−x0ynxn−x0.Lemma 4 (see [14–16]).
SupposeF is a set of continuous functions which satisfy f:x0,xn⟶R and fx0=y0andfxn=yn. The metric is defined by the following formula:(10)df,g=maxfx−gx,x∈x0,xn,∀f,g∈F.Then,F,d is a complete metric space. Let the real numbers ai,ci,ei,andfi, be defined by equations (5)–(9). Define a mapping T:F⟶F by(11)Tfx=ciLi−1x+difLi−1x+fi,x∈xi−1,xi,i=1,2,…,n,where Li:x0,xn⟶xi−1,xi is the invertible transformation:(12)Lix=aix+ei,(13)Li−1x=x−eiai,Li−1xi−1=x0,Li−1xi=xn.Then, Tf is continuous on the interval xi−1,xi and T is a contraction mapping on F,d, so T possesses a unique fixed point in F. That is, there exists a function f∈F such that(14)Tf=f,∀f∈F.The functionf is called fractal interpolation function. The abbreviation “FIF” is used for “fractal interpolation function.”


## 3. The Continuity and the Uniform Continuity of Fractal Interpolation Function

Theorem 1.
Let functionf be a FIF generated by IFS mentioned in Definition 4 and defined by affine mapping referred from equations (3)–(14) and Lemmas 3 and 4. Then, the FIF is continuous on the closed interval on the closed interval x0,xn.Proof.
∀x¯∈x0,xn, according to Lemmas 3 and 4 and equations (11) to (14), ∀ε>0, and(15)|fx−fx¯=|Tfx−Tfx¯=ciLi−1x+difLi−1x+fi−ciLi−1x¯−difLi−1x¯−fi=ciLi−1x−Li−1x¯+difLi−1x−fLi−1x¯=cix−eiai−x¯−eiai+difx−eiai−fx¯−eiai=ciaix−x¯+dicix+dix−eiai+fi−cix¯−dix¯−eiai−fi=ci+di2ai+cidix−x¯<ε.
Select(16)p=max1≤i≤nci+di2ai+cidi.Then, we select(17)δ=εp,so ∀ε>0, for any x∈x0,xn and x−x¯<δ, such that(18)fx−fx¯<ε.Therefore, according to Definition 1, fractal interpolation function f is continuous on the point x¯. Furthermore, because x¯ is arbitrarily selected from x0,xn and f is a continuous function on x0,xn.
The proof above illustrates that fractal interpolation functionf is continuous on the closed interval x0,xn.Theorem 2.
Let functionf be a FIF generated by IFS and defined by affine transformation from equations (4)–(14) on the closed interval x0,xn. Then, f is uniformly continuous on the closed interval x0,xn.Proof.
The contradiction method will be used in the proof. Suppose that FIF is not uniformly continuous on the closed intervalx0,xn. According to Definition 2,(19)∃ε0>0,∀δ>0,∃xm′,xm″∈x0,xn,so that(20)limm⟶∞xm′−xm″=0,that is,(21)xm′−xm″<δ,but(22)fxm′−fxm″≥ε0.
Becausexm′∈x0,xn, the number sequence xm′ is a bounded number sequence. According to Lemma 1, xm′ has a convergent subsequence xmk′. Let xmk″⟶x¯; then,(23)limk⟶∞xmk″=limk⟶∞xmk″−xmk′+xmk′=limk⟶∞xmkx=x¯,from x0≤xmk′≤xn⟹x0≤x¯≤xn. Because FIF is a continuous function on the closed interval x0,xn,(24)limx⟶x¯fx=fx¯.
From the relationship between limit of function and that of number sequence,(25)limk⟶∞fxmk′−fxmk″≥ε0,fx¯−fx¯≥ε0,that is,(26)0≥ε0,which is contrary to ε0>0. In other words, at first, the contrary hypothesis to the conclusion of Theorem 2 makes ε0>0 and ε0≤0 contradictory. So, f is uniform continuous function on the interval x0,xn.
For the proof of Theorem2, we have the following second method to prove Theorem 2.Proof.
Because fractal interpolation functionf is continuous on the interval x0,xn, ∀x¯∈x0,xn,∀ε>0,∃δx¯>0, so that ∀x∈∪x¯;δx¯ and(27)fx−fx¯<ε/2.
Let(28)H=∪x¯;δx¯2|x¯∈x0,xn.Obviously, H is an open covering of x0,xn. According to Lemma 2 above, there exists(29)H∗=∪x¯i;δi2|i=1,2,…,N,to cover x0,xn. Suppose that(30)δ=min1≤i≤Nδi2,∀x′,x″∈x0,xn, and x′−x″<δ; then, ∃∪x¯i;δi/2∈H∗, such that(31)x′∈∪x¯i;δi2.
On the one hand,(32)x′−x¯i<δi2<δi⟹fx′−fx¯i<ε2.On the other hand,(33)x″−x¯i=x″−x′+x′−x¯i<δ+δi2<δi⟹fx″−fx¯i<ε2.
Therefore,(34)fx′−fx″=fx′−fx¯i+fx¯ifx″<ε2+ε2=ε.
According to Definition2, fractal interpolation function is a uniformly continuous function proved by finite covering Lemma 2.
In other words, from the two proof methods of Theorem2, fractal interpolation function f is uniform continuous on the interval x0,xn.


## 4. Conclusion

From the above related theory of fractal interpolation function and function continuity, the following conclusion can be acquired through the above proof. Fractal interpolation function is not only continuous but also uniformly continuous on the closed intervalx0,xn.



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