A fractal interpolation scheme for a possible sizeable set of data

Abstract

In this paper, we propose a new fractal interpolation scheme. More precisely, we consider $a,b\in \mathbb{R}$, $a<b$, and $A\subseteq \mathbb{R}$ such that $\{a,b\}\subseteq A=\overline{A}\subseteq [a,b]$ and $\stackrel{\circ }{A}=\varnothing$ and prove that for every continuous function $f:A\to \mathbb{R}$, there exist a continuous function $g^{\ast }:[a,b]\to \mathbb{R}$ such that $g_{\mid A}^{\ast }=f$ and a possible infinite iterated function system whose attractor is the graph of $g^{\ast }$. If $A$ is finite, we obtain the classical Barnsley’s interpolation scheme and for $A=\{x_n\mid n\in \mathbb{N}\}\cup \{b\}$, where $x_1=a$, ${\lim }_{n\to \infty }{x}_n=b$ and $x_n\in [a,b]$ for every $n\in \mathbb{N}$, we obtain a countable scheme due to N. Secelean. Our interpolation scheme permits $A$ to be uncountable as it is the case for the Cantor ternary set.