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 a fractal interpolation function. | Turk J Math (2017) 41: 829 – 840 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Graph-directed fractal interpolation functions ˙ ¨ ˙ ∗ Ali DENIZ, Yunus OZDEM IR Department of Mathematics, Anadolu University, Eski¸sehir, Turkey Received: • Accepted/Published Online: • Final Version: 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 a fractal interpolation function. We generalize the notion of the 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 a corresponding data set in R2 such that the graphs of the interpolation functions are attractors of a graph-directed iterated function system. Key words: Fractal interpolation function, iterated function system, graph-directed iterated function system 1. Introduction Barnsley introduced fractal interpolation functions (FIF) using the iterated function system (IFS) theory, which is an important part of fractals (see [1, 2, 13] for further information). He showed that there exists a function interpolating a given data set such that the graph of this function is the attractor of an IFS. In recent decades it has been widely used in various fields such as approximation theory, image compression, modeling of signals, and many other scientific areas ([4, 8, 11, 12, 14]). Let us first summarize the notions of IFS and FIF. An IFS is a finite collection of contraction mappings φi : X → X (i = 1, ., n) on a complete metric space X . It is known that there exists a unique nonempty compact set A satisfying A= n ∪ φi (A) ⊂ X, i=1 which is called the attractor of the IFS (see [7]). A data set is a set of points D = {(x0 , F0 ), (x1 , F1 ), . . . , (xN , FN )} ⊂ .
