Graph-Directed Fractal Interpolation Functions

TL;DR

This paper extends fractal interpolation functions to the graph-directed case, demonstrating that multiple data sets can be interpolated simultaneously with their graphs forming attractors of a graph-directed iterated function system.

Contribution

It generalizes the concept of fractal interpolation functions to the graph-directed setting, establishing existence results for multiple data sets.

Findings

Existence of graph-directed fractal interpolation functions for multiple data sets

Graphs of these functions are attractors of graph-directed iterated function systems

Generalization from single to multiple data set interpolation

Abstract

It is known that there exists a function interpolating a given data set such that the graph of the function is the attractor of an iterated function system which is called fractal interpolation function. We generalize the notion of fractal interpolation function to the graph-directed case and prove that for a finite number of data sets there exist interpolation functions each of which interpolates corresponding data set in $\mathbb{R}^2$ such that the graphs of the interpolation functions are attractors of a graph-directed iterated function system.