1 Introduction and preliminaries
The fractal dimension is one of the major themes in Fractal Geometry. Estimation of the fractal dimension of sets and graphs has received much attention (see [Reference Barnsley5, Reference Barnsley6, Reference Falconer10]). The study of dimensions of graphs began with the Hausdorff dimension of Weierstrass-type functions (see [Reference Hunt12, Reference Mauldin and Williams17]). In [Reference Mauldin and Williams17], Mauldin and Williams considered such a function,
$$ \begin{align*}W_b(x)=\sum_{n=-\infty}^\infty b^{-\alpha n} [\Phi (b^nx+\theta_n)-\Phi(\theta_n)],\end{align*} $$
where
$b>1,~ 0<\alpha <1$
,
$\Phi $
has period one and
$\theta _n$
is an arbitrary number, and established results on the Hausdorff dimension when the function satisfies a convex-Lipschitz condition. This is the major motivation for our work. By using the definition of a convex Lipschitz function, we introduce the convex-Lipschitz space and estimate the Hausdorff dimension and box dimension of a general fractal interpolation function (FIF).
The concept of FIF was introduced by Barnsley [Reference Barnsley5] using the notion of an iterated function system (IFS). Recent related work on dimension theory can be seen in [Reference Bárány, Rams and Simon3, Reference Bárány, Rams and Simon4]. In [Reference Bárány, Rams and Simon3], Bárány et al. applied a result of Hochman [Reference Hochman11] on self-similar sets with overlaps, to compute the Hausdorff dimension of self-affine sets. They also studied the dimension theory of diagonally homogeneous triangular planar self-affine sets in [Reference Bárány, Rams and Simon4].
1.1 Fractal interpolation functions
We outline the construction of FIF and refer to [Reference Barnsley5, Reference Barnsley6] for the details.
Assume that
$ \{(x_n,y_n) : n=1,2,\ldots ,N\} $
is a set of interpolation points. We write
$I= [x_1, x_N] $
and
$J = \{1,2,\ldots ,N-1\}$
, and let
$I_j= [x_j, x_{j+1}]$
for
$ j \in J$
. Let
$L_j:I \rightarrow I_j$
,
$ j\in J$
, be contractive homeomorphisms with
$$ \begin{align*} L_j(x_1)=x_j, \quad L_j(x_N)=x_{j+1}, \quad j \in J.\end{align*} $$
Let
$F_j: I\times \mathbb {R} \rightarrow \mathbb {R}$
be a mapping satisfying, for
$j \in J$
,
$r_j \in [0,1)$
:
-
(i)
$ F_j(x_1,y_1)=y_j, F_j(x_N,y_N)=y_{j+1}$
; -
(ii)
$ |F_j(x,y) - F_j(x,y')| \,\leq \, r_j |y- y'|$
for all
$x\in I$
and
$y,y' \in \mathbb {R}.$
We consider
$$ \begin{align*}L_j(x)=a_j x+ b_j, \quad F_j(x,y) = \alpha_j y + q_j (x).\end{align*} $$
Here,
$a_j$
and
$b_j$
can be determined by using the conditions
$ L_j(x_1)=x_j, L_j(x_N)=x_{j+1}$
. The scaling factor
$\alpha _j$
satisfies
$-1< \alpha _j <1$
and we set
$|\alpha _j|_\infty =\max _j \{\alpha _j\}.$
The ‘join-up conditions’, which are imposed on the maps
$F_j$
, are given by
$q_j(x_1)=y_j-\alpha _j y_1$
and
${q_j(x_N)=y_{j+1}-\alpha _j y_N}$
for all
$j \in J $
for suitable continuous functions
$q_j:I \to \mathbb {R}.$
Let us define
$W_j:I\times \mathbb {R} \rightarrow I\times \mathbb {R} $
for
$j \in J$
by
$$ \begin{align*}W_j(x,y)=(L_j(x),F_j(x,y)). \end{align*} $$
Then
$\mathcal {I}:=\{I\times \mathbb {R};W_1,W_2,\ldots ,W_{N-1}\}$
is the IFS. Barnsley [Reference Barnsley5] proved that
$\mathcal {I}$
has a unique invariant set which is the graph of a continuous function
$f:I \to \mathbb {R},$
referred to as a FIF, and that it satisfies the self-referential equation
$$ \begin{align*}f(x)= \alpha_j f (L_j^{-1}(x))+ q_j (L_j^{-1}(x)), \quad x \in I_j, j \in J.\end{align*} $$
There are various approaches to fractal dimensions of fractal functions. These include the use of the mass-distribution principle, potential theory, Fourier transforms and positive operators to compute or estimate the Hausdorff dimension of a set [Reference Falconer10, Reference Nussbaum, Priyadarshi and Verduyn Lunel18, Reference Verma24]. Using the potential theoretic approach, Barnsley [Reference Barnsley5] gave results on the Hausdorff dimension of an affine FIF. Falconer [Reference Falconer10] also gave estimates for the Hausdorff dimension of an affine FIF. Results on the Hausdorff dimension using the positive operators approach are given in [Reference Nussbaum, Priyadarshi and Verduyn Lunel18, Reference Verma24]. This approach is used to discuss the continuity of the Hausdorff dimension of the invariant set in [Reference Priyadarshi20].
Pandey et al. [Reference Pandey, Som and Verma19] considered the fractal dimension for set valued mappings using the
$\delta $
-covering method. Jha and Verma [Reference Jha and Verma13] gave results for the fractal dimensions of fractal functions and some invariant sets. They estimated fractal dimensions for a class of FIFs, known as
$\alpha $
-fractal functions. Agrawal and Som [Reference Agrawal and Som1, Reference Agrawal and Som2] gave results for
$\alpha $
-fractal functions on Sierpiński gaskets. Sahu and Priyadarshi [Reference Sahu and Priyadarshi23] estimated the box dimension of the graph of harmonic functions on Sierpiński gaskets.
Ruan et al. [Reference Ruan, Su and Yao21] estimated the box dimension of a new class of linear FIFs by using the
$\delta $
-covering method. Additionally, they established a relationship between the order of a fractional integral and box dimensions of two linear FIFs. Estimates of the box dimension of bilinear fractal interpolation surfaces are given in [Reference Kong, Ruan and Zhang14]. A recurrent FIF is the generalisation of a linear FIF and the graph of a recurrent FIF is the invariant set of a recurrent IFS. Ruan et al. [Reference Ruan, Xiao and Yang22] gave the construction of a recurrent FIF under certain assumptions and estimated the box dimension of the self-affine recurrent FIFs.
Work on the fractal dimension of fractional integrals can be seen as a connection between fractal geometry and fractional calculus. The bounded variation property of a continuous function plays a significant role in estimating the box dimension. Using this approach, Liang [Reference Liang15] gave interesting results on the box dimension of the Riemann–Liouville fractional integral. He proved that if a function f is continuous and of bounded variation on
$[0,1],$
then
$\dim _B(f)=1$
and the box dimension of the Riemann–Liouville fractional integral corresponding to f is also
$=1$
[Reference Liang16]. Liang estimated the exact box dimension of the Riemann–Liouville fractional integral of one-dimensional continuous functions. We gave the fractal dimension of the mixed Riemann–Liouville fractional integral on a rectangular region in [Reference Chandra and Abbas8] and estimated fractal dimensions for various choices of continuous functions such as Hölder continuous function, functions having box dimension two and unbounded variational continuous functions.
In Sections 2 and 3, we give dimension results on convex-Lipschitz space and oscillation space, respectively.
1.2 Definitions
We complete Section 1 with some definitions and terminologies. For further definitions related to the fractal dimension, we refer to [Reference Falconer10].
Let
$F \neq \emptyset $
be a subset of
$\mathbb {R}^n$
. The diameter of F is given by
$$ \begin{align*}\lvert F \rvert=\sup \{\lVert x-y \rVert_2:x,y\in F\}.\end{align*} $$
If
$\{F_i\}$
is a countable (or finite) collection of sets having diameter at most
$\delta $
which cover the set
$E\subseteq \mathbb {R}^n,$
then we say that
$\{F_i\}$
is a
$\delta $
-cover of
$E.$
For
$\delta>0$
and a nonnegative real number s, we define
$$ \begin{align} H_\delta^s(E)=\inf \bigg \{\sum_{i=1}^\infty |F_i|^s: \{F_i\} ~\text{is a}~ \delta\text{-cover of}~ E\bigg \}. \end{align} $$
Definition 1.1. The s-dimensional Hausdorff measure of E is
$H^s(E)=\lim _{\delta \to 0} H_\delta ^s(E).$
Definition 1.2 (Hausdorff dimension).
Let
$s\ge 0$
and
$E\subseteq \mathbb {R}^n.$
The Hausdorff dimension of E is defined as
$$ \begin{align*}\dim_H(E)=\inf \{s:H^s(E)=0 \}=\sup \{s:H^s(E)=\infty\}.\end{align*} $$
Definition 1.3 (Box dimension).
Let
$E \subseteq \mathbb {R}^n$
be bounded and nonempty and let
$N_{\delta }(E)$
be the smallest number of sets of diameter at most
$\delta $
which cover
$E.$
The lower box dimension of E is
$$ \begin{align*}\underline{\dim}_B(E)=\liminf_{\delta \to 0} \frac{\log N_{\delta}(E)}{- \log \delta}\end{align*} $$
and the upper box dimension of E is
$$ \begin{align*}\overline{\dim}_B(E)={\limsup_{\delta \to 0}} \frac{\log N_{\delta}(E)}{- \log \delta}.\end{align*} $$
If both, lower and upper box dimensions are the same, then that quantity is called the box dimension of E and it is given by
$$ \begin{align*}\dim_B(E)=\lim_{\delta \rightarrow 0} \frac{\log N_{\delta}(E)}{- \log \delta}.\end{align*} $$
2 Convex-Lipschitz space
In this section, first we show that fractal functions associated with some IFS belong to the class of convex Lipschitz functions. Then we estimate the Hausdorff dimension and the box dimension of fractal functions in this class.
Definition 2.1 [Reference Mauldin and Williams17].
Let
$\theta : \mathbb {R}^+ \to \mathbb {R}^+$
. A function f is called convex Lipschitz of order
$\theta $
on an interval
$[a,b]$
provided there exists a constant M such that
$$ \begin{align*}|\Delta(x,y,\delta)|:=|f(x+\delta y)-(\delta f(x+y)+(1-\delta)f(x))|\le M \theta(y),\end{align*} $$
for
$a\le x<x+y \le b$
and
$0 \le \delta \le 1$
. The convex-Lipschitz space of order
$\theta $
is
$$ \begin{align*}V^\theta (I)=\{f:I \to \mathbb{R}:f~ \text{is convex Lipschitz of order} ~\theta\}.\end{align*} $$
It can be seen that
$V^\theta $
is a vector space over the field
$\mathbb {R}$
. For
$f\in V^\theta (I)$
, we define
$\lVert f \rVert _{V^\theta }=\lVert f \rVert _\infty +[f]^*,$
where
$$ \begin{align*}[f]^*=\sup_{a\le x<x+y \le b}\frac{|f(x+\delta y)-(\delta f(x+y)+(1-\delta)f(x))|}{\theta(y)}.\end{align*} $$
It is easy to check that
$\lVert \cdot \rVert _{V^\theta }$
defines a norm on
$V^\theta (I)$
.
Lemma 2.2. If
$f:I \to \mathbb {R}$
and
$(f_k)$
is a sequence of continuous functions which converges uniformly to f, then
$[f_n-f]^* \le \liminf _{k\rightarrow \infty } [f_n-f_k]^*$
.
Proof. By using the triangle inequality,
$$ \begin{align*} & \frac{|f_n(x+\delta y)-f(x+\delta y)-[\delta f_n(x+y)-\delta f(x+y)+(1-\delta)f_n(x)-(1-\delta)f(x)]|}{\theta(y)} \\[-1pt] &\quad = \lim_{k \to \infty} \frac{1}{\theta(y)} \{|f_n(x+\delta y)-f_k(x+\delta y) \\[-1pt] & \ \ \,\qquad\qquad\quad - [\delta f_n(x+y)-\delta f_k(x+y)+(1-\delta)f_n(x)-(1-\delta)f_k(x)]| \} \\[-1pt] &\quad \le \liminf_{k \to \infty} \sup_{a\le x<x+y \le b} \frac{1}{\theta(y)} \{|f_n(x+\delta y)- f_k(x+\delta y) \\[-1pt] & \, \qquad\qquad\qquad\qquad\qquad -[\delta f_n(x+y)-\delta f_k(x+y)+(1-\delta)f_n(x)-(1-\delta)f_k(x)]| \}. \end{align*} $$
This completes the proof.
Theorem 2.3. The space
$(V^\theta (I),\lVert \cdot \rVert _{V^\theta })$
is a Banach space.
Proof. Let
$(f_n)$
be a Cauchy sequence with respect to
$\lVert \cdot \rVert _{V^\theta }$
in
$V^\theta (I)$
. This means that for any
$\epsilon>0,$
there exists a natural number
$n_0$
such that
$\lVert f_n-f_k \rVert _{V^\theta } < \epsilon $
for all
$n,k \ge n_0$
.
From the definition of the norm
$\lVert \cdot \rVert _{V^\theta }$
, it follows that
$\lVert f_n-f_k \rVert _\infty < \epsilon $
for all
${n,k \ge n_0.}$
Because
$(C(I),\lVert \cdot \rVert _\infty )$
is a Banach space, there is a continuous function f such that
${\Vert f_n-f \rVert _\infty \to ~0}$
as
$n\to \infty .$
We claim that
$f \in V^\theta (I)$
and
$\lVert f_n-f\rVert _{V^\theta } \to 0$
as
$n \to \infty .$
Let
$n\ge n_0.$
In view of Lemma 2.2,
$$ \begin{align*} \|f_n-f\|_{V^\theta} = \|f_n-f\|_\infty + [f_n-f]^* & \le \liminf_{k\rightarrow\infty} \{\|f_n-f_k\|_\infty + [f_n-f_k]^*\} \\ & \le \sup_{k\ge n_0} \|f_n-f_k\|_{v^\theta} \le \epsilon. \end{align*} $$
Hence, we obtain
$f-f_{n_0} \in V^\theta (I)$
. Consequently,
$f=f-f_{n_0}+ f_{n_0} \in V^\theta (I)$
and we have
$\lVert f_n-f\rVert _{V^\theta } \le \epsilon $
for all
$n \ge n_0$
. This completes the proof.
Definition 2.4 [Reference Falconer10, Section 2.5].
Let
$ E \subset \mathbb {R}^n$
and suppose that the dimension function
$\theta : \mathbb {R}^+ \to \mathbb {R}^+$
is increasing and continuous. Analogously to (1.1), we define
$$ \begin{align*} H_\delta^\theta(E)=\inf \bigg \{\sum \theta (|F_i|): \{F_i\} ~\text{is a}~ \delta\text{-cover of}~ E\bigg \}.\end{align*} $$
This leads to a measure, by taking
$H^\theta (E)=\lim _{\delta \to 0} H_\delta ^\theta (E).$
If
$\theta (t)=t^s$
, it is the usual definition of an s-dimensional Hausdorff measure.
Theorem 2.5 [Reference Mauldin and Williams17].
Let
$\theta : \mathbb {R}^+ \to \mathbb {R}^+$
be a continuous map such that:
-
(i)
$\theta (t)>0$
for
$t>0$
; -
(ii)
$\limsup _{t \to 0} {t}/{\theta (t)}< \infty $
and -
(iii) there is a
$\beta \ge 0$
such that
$\lim _{t \to 0}{ \theta (ct)}/{\theta (t)}=c^\beta $
for all
$c>0.$
If f is a continuous map on
$[0,1]$
and also convex Lipschitz of order
$\theta $
, then f has
$\sigma $
-finite h measure, where
$h(y)={y^2}/{\theta (y)}.$
Theorem 2.6. Under the hypotheses of the above theorem:
-
• if
$\theta (y)=y^\alpha $
then
$\dim _H({\mathrm {Graph}}(f)) \le \overline {\dim }_B({\mathrm {Graph}}(f))\le 2-\alpha $
; -
• if
$\theta (y)=y \ln (1/y)$
then
$\dim _H({\mathrm {Graph}}(f))= \dim _B({\mathrm {Graph}}(f))=1$
.
Proof. The results follow from Theorem 2.5 and the definitions of the Hausdorff measure and Hausdorff dimension.
Theorem 2.7. Let
$q_j \in V^\theta (I)$
and
$\alpha _j \in (-1,1).$
Then the associated fractal interpolation function f is in
$V^\theta (I)$
provided that
$\max \{|\alpha _j|_\infty ,\max _j |\alpha _j|{\theta (Y)}/{\theta (a_jY)}\}<1$
.
Proof. We first define
$V_*^\theta (I):=\{f \in V^\theta :f(x_1)=y_1,~f(x_N)=y_N\}$
. Since
$V_*^\theta (I)$
is a closed subset of
$V^\theta (I)$
, it follows that
$V_*^\theta (I)$
is a complete metric space with respect to the metric induced by the norm
$\| \cdot \|_{V^\theta }$
. Let us define a map
$T:V_*^\theta (I) \to V_*^\theta (I)$
by
$$ \begin{align*}(Tf)(x)=\alpha_j f (L_j^{-1}(x))+ q_j (L_j^{-1}(x)), \quad x \in I_j, j \in J.\end{align*} $$
Here,
$L_j(x) = a_jx+b_j$
and
$L_j^{-1}(x) = x/a_j-b_j$
. Set
$X={x}/{a_j}-b_j$
and
$Y={y}/{a_j}$
. The mapping T is well defined and, for
$f,g \in V_*^\theta (I)$
,
$$ \begin{align*} &\lVert Tf-Tg \rVert_{V^\theta}= \lVert Tf-Tg \rVert_\infty +[Tf-Tg]^* \\ &\quad\le |\alpha_j|_\infty \|f(X)-g(X)\|_\infty + \max_j \sup_{a\le a_j(X+b_j)<a_j(X+Y+b_j) \le b} \\ &\qquad\qquad\quad \frac{|\alpha_j||(f-g)( X+\delta Y)-(\delta (f-g)( X+Y)+(1-\delta)(f-g)(X)|}{\theta (Y)}\times \frac{\theta(Y)}{\theta (a_jY)}\\ &\quad\le |\alpha_j|_\infty \|f-g\|_\infty + \max_j |\alpha_j|\frac{\theta (Y)}{\theta (a_j Y)} [f-g]^* \\ &\quad\le \max \bigg\{|\alpha_j|_\infty,\max_j |\alpha_j|\frac{\theta(Y)}{\theta(a_jY)}\bigg\} \lVert f-g \rVert_{V^\theta}. \end{align*} $$
Since
$\max \{|\alpha _j|_\infty ,\max _j |\alpha _j|{\theta (Y)}/{\theta (a_jY)}\}<1$
, the mapping T is a contraction on
$V_*^\theta (I)$
. From the Banach fixed point theorem, T has a unique fixed point
$f \in V_*^\theta (I)$
. From
$T(f)=f$
, we can write
$$ \begin{align} f(L_j(x))= \alpha_j f(L_j(x))+q_j(x) \quad \mbox{for } x \in I, j \in J. \end{align} $$
For
$j \in J := \{1,2,3, \ldots , N-1\}$
, let us define
$W_j: I \times \mathbb {R} \rightarrow I \times \mathbb {R} $
by
$$ \begin{align} W_j(x,y) = (L_j(x), \alpha_j y + q_j (x)). \end{align} $$
We have shown above that the graph of f is an attractor of the IFS
$\{I \times \mathbb {R}; W_j,j \in J\}$
. By using the proof of Theorem 1 in [Reference Barnsley5], we can show that the attractor associated with this IFS is the graph of f. In fact, it is the graph of the fractal perturbation of f. To see this, we take the functional equation (2.1), the definition of
$W_j$
from (2.2) and
$I = \bigcup _{j \in J} L_j(I)$
, and get
$$ \begin{align*} \begin{aligned} \bigcup_{j \in J} W_j({\mathrm{Graph}}(f)) & = \bigcup_{j \in J} \{(L_j(x),f(L_j(x))):x \in I \}\\ & = \bigcup_{j \in J} \{(x,f(x)):x \in L_j(I) \} = {\mathrm{Graph}}(f), \end{aligned} \end{align*} $$
completing the proof.
By combining Theorems 2.6 and 2.7, we can estimate the fractal dimension of certain fractal interpolation functions.
Theorem 2.8. Let
$q_j \in V^\theta (I)$
and
$\alpha _j \in (-1,1)$
be such that
$$ \begin{align*}\textstyle \max \{|\alpha_j|_\infty,\max_j |\alpha_j|{\theta(Y)}/{\theta(a_jY)}\}<1.\end{align*} $$
Then we have the following bounds for the fractal dimension of the graph of the associated fractal interpolation function f.
-
• If
$\theta (y)=y^\alpha $
, then
$\dim _H({\mathrm {Graph}}(f)) \le \overline {\dim }_B({\mathrm {Graph}}(f))\le 2-\alpha $
. -
• If
$\theta (y)=y \ln (1/y)$
, then
$\dim _H({\mathrm {Graph}}(f))= \dim _B({\mathrm {Graph}}(f))=1$
.
Example 2.9 (Weierstrass-type function).
For more details on this example, we refer to [Reference Mauldin and Williams17]. Let
$\Phi : \mathbb {R} \to \mathbb {R}$
be a bounded function which is convex Lipschitz of order 1. For
${b>1,~0<\alpha <1}$
, define
$$ \begin{align*}f(x)=\sum_{n=0}^\infty b^{-\alpha n} \Phi (b^nx+\theta_n).\end{align*} $$
Then f is convex Lipschitz of order
$\alpha .$
Consequently,
$\dim _H({\mathrm {Graph}}(f)) \le 2-\alpha $
. If
${\alpha =1}$
, then
$\dim _H({\mathrm {Graph}}(f)) =1.$
Remark 2.10. Note that any continuous function
$f:[a,b] \to \mathbb {R}$
satisfies the convex-Lipschitz condition with
$\theta (y)=$
constant. For any continuous function f, we have
$1 \le \dim ({\mathrm {Graph}}(f)) \le 2$
. So, for constant
$\theta $
, we cannot conclude any nontrivial dimension estimates.
3 Oscillation spaces
We refer to [Reference Carvalho7, Reference Deliu and Jawerth9] for more details on oscillation spaces. Let
$Q \subset [0,1]$
be a p-adic subinterval, that is,
$ Q =[\,jp^{-m},(j+1)p^{-m}]$
for some integers j and m with
$m \ge 0$
and
$0 \le j < p^{-m}$
. The oscillation of a continuous function
$g:[0,1] \to \mathbb {R}$
over Q is given by
$$ \begin{align*} \mathcal{R}_g(Q)= \sup_{x_1,x_2 \in Q} |g(x_1) -g(x_2)| = \sup_{x_1 \in Q}g(x_1) - \inf_{x_2 \in Q} g(x_2), \end{align*} $$
and the total oscillation of order m is given by
$$ \begin{align*} {\mathrm{Osc}}(m,g)= \sum_{|Q|=p^{-m}} \mathcal{R}_g(Q),\end{align*} $$
where the sum is taken over all p-adic intervals
$ Q \subset [0,1]$
having length
$|Q|=p^{-m}.$
The oscillation space
$V^{\beta }(I),~\beta \in \mathbb {R}$
, is defined by
$$ \begin{align*} V^{\beta}(I)= \bigg\{g \in \mathcal{C}(I): \sup_{m \in \mathbb{N} }\frac{ {\mathrm{Osc}}(m,g)}{p^{m(1-\beta)}}< \infty \bigg\}.\end{align*} $$
We also define
$$ \begin{align*} V^{\beta-}(I)= \{g \in \mathcal{C}(I): g \in V^{\beta- \epsilon}(I) \mbox{ for all } \epsilon>0 \}\end{align*} $$
and
$$ \begin{align*} V^{\beta+}(I)= \{g \in \mathcal{C}(I): g \notin V^{\beta+ \epsilon}(I) \mbox{ for all } \epsilon>0 \}.\end{align*} $$
Theorem 3.1 [Reference Carvalho7, Theorem 4.1]; see also [Reference Deliu and Jawerth9].
For a real-valued continuous function g which is defined on I and
$0<\beta \le 1$
,
$$ \begin{align*}\overline{\dim}_B({\mathrm{Graph}}(g)) \le 2 - \beta \quad\mbox{if and only if}\ g \in V^{\beta-}(I) \end{align*} $$
and
$$ \begin{align*}\overline{\dim}_B({\mathrm{Graph}}(g)) \ge 2 - \beta \quad\mbox{if and only if}\ g \in V^{\beta+}(I). \end{align*} $$
Theorem 3.2. Let
$q_j \in V^{\beta }(I)$
,
$\alpha _j \in (-1,1)$
and
$\max \{ |\alpha _j|_\infty , \sum _{j \in J}|\alpha _j|_{\infty } \}<1$
. Then the fractal interpolation function
$f \in V^{\beta }(I).$
Moreover,
$\overline {\dim }_B({\mathrm {Graph}}(f)) \le 2 - \beta .$
Proof. Let
$V^{\beta }_*(I)= \{ f \in V^{\beta }(I ): f(x_1)=y_1, ~f(x_N)=y_N \}.$
It can be seen that the space
$V^{\beta }_*$
is a closed subset of
$V^{\beta }(I).$
It follows that
$V_*^\beta (I)$
is a complete metric space with respect to the metric induced by the norm
$\| \cdot \|_{V^\beta }$
. Let us define a map
$ T: V^{\beta }_*(I) \rightarrow V^{\beta }_*(I) $
by
$$ \begin{align*}(Tf)(x)=\alpha_j f (L_j^{-1}(x) )+ q_j (L_j^{-1}(x)), \quad x \in I_j, j \in J.\end{align*} $$
Set
$X={x}/{a_j}-b_j$
, so that
$L_j^{-1}(x)=X$
. Then T is well defined and, for
$g, h \in V^{\beta }_*(I )$
,
$$ \begin{align*} \|Tf -Tg\|_{V^{\beta}} & = \|Tf -Tg\|_{\infty} + \sup_{m \in \mathbb{N}}\frac{{\mathrm{Osc}}(m,Tf-Tg)}{p^{m (1-\beta)}}\\ & \le \lvert \alpha_j \rvert_\infty \lVert f(X)-g(X)\rVert _\infty+ \sum_{j \in J}|\alpha_j|_{\infty} \sup_{m \in \mathbb{N}}\frac{{\mathrm{Osc}}(m,f(X)-g(X))}{p^{m(1-\beta)}} \\ & \le \max \bigg \{ |\alpha_j|_\infty, \sum_{j \in J}|\alpha_j|_{\infty} \bigg \} \|f-g\|_{V^{\beta}}. \end{align*} $$
Since
$\max \{ |\alpha _j|_\infty , \sum _{j \in J}|\alpha _j|_{\infty } \}<1$
, the mapping T is a contraction on
$ V^{\beta }_*(I).$
From the Banach fixed point principle, T has a unique fixed point
$f \in V^{\beta }_*(I)$
, completing the proof.
3.1 Graphs of fractal interpolation functions
Figures 1–4 give approximate graphs of some fractal interpolation functions.
Plot for
$\alpha =0.0$
.
Plot for
$\alpha =0.3$
.
Plot for
$\alpha =0.6$
.
Plot for
$\alpha =0.9$
.
For this example, let
$g=19+8\cos (3x)$
and
$q(x)=(1-\alpha (1+x^2-x))\cdot g(x), x\in [0,1].$
We show the graphs of the fractal interpolation function f for scaling factors
$\alpha = 0.0,~ 0.3, 0.6,~0.9$
in Figures 1–4 respectively.
Acknowledgements
We are thankful to the anonymous reviewers and handling editor for their comments and suggestions, which helped us to improve the manuscript.
