# A fractal interpolation scheme for a possible sizeable set of data

### Radu Miculescu

Transilvania University of Brașov, Romania### Alexandru Mihail

University of Bucharest, Romania### Cristina Maria Pacurar

Transilvania University of Brașov, Romania

## Abstract

In this paper, we propose a new fractal interpolation scheme. More precisely, we consider $a,b∈R$, $a<b$, and $A⊆R$ such that ${a,b}⊆A=A⊆[a,b]$ and $A∘=∅$ and prove that for every continuous function $f:A→R$, there exist a continuous function $g_{∗}:[a,b]→R$ such that $g_{∣A}=f$ and a possible infinite iterated function system whose attractor is the graph of $g_{∗}$. If $A$ is finite, we obtain the classical Barnsley’s interpolation scheme and for $A={x_{n}∣n∈N}∪{b}$, where $x_{1}=a$, $lim_{n→∞}x_{n}=b$ and $x_{n}∈[a,b]$ for every $n∈N$, 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.

## Cite this article

Radu Miculescu, Alexandru Mihail, Cristina Maria Pacurar, A fractal interpolation scheme for a possible sizeable set of data. J. Fractal Geom. 9 (2022), no. 3/4, pp. 337–355

DOI 10.4171/JFG/117