Fractal interpolation functions

One of the most important tools of fractal geometry is iterated function systems. These systems, introduced in a study published by Hutchinson in 1981, allow many sets to be modeled mathematically. Most fractals are also expressed as the attractor of an iterated function system. Finding a continuous function that interpolates a given finite set of points in an Euclidean space is an important and currently studied problem. The existence of continuous functions that interpolate the data set given in the plane and whose graph is the attractor of an iterative function system was first studied and demonstrated in the 1980s. It is extremely important since it is a problem with a wide application area. Therefore, it is of course quite meaningful to seek an answer to this question in different spaces, as well as in higher dimensional spaces. In later studies, the existence of a continuous function that interpolates a given data set in three dimensional Euclidean space has been shown under some constraints regarding the given data set, with its graph being the attractor of an iterated function system. These constraints have been stretched over time and the existence of sets (graphs of a fractal interpolation functions) called fractal interpolation surfaces has been investigated for a much more flexible data sets. In the first two sections of this thesis, the notions of iterated function systems and (classical) fractal interpolation functions were introduced. Then, studies on fractal interpolation surfaces were summarized and several important studies were explained and exemplified in details. In addition, the studies and the results obtained on the fractal dimensions of the fractal interpolation surfaces obtained were also investigated and presented in the related sections.