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 \overset{\circ}{A}=\emptyset and prove that for every continuous function f:A\rightarrow\mathbb{R} , there exist a continuous function g^{\ast}:[a,b]\rightarrow \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\rightarrow\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.