Chapter 5 - Fractal Function Construction

Abstract

This chapter discusses a special class of continuous functions. These functions are referred to as fractal functions, because their graphs usually have nonintegral dimension. These fractal functions may be used for interpolation and approximation purposes and are in this way analogous to splines. The chapter presents a general construction based on a Read-Bajraktarevic operator acting on L∞(X, Y). The Read-Bajraktarevic operator provides a natural framework for the description of fractal functions in terms of hyperbolic iterated function systems (IFSs). The chapter presents several other related constructions and they are set into perspective. It also discusses Peano curves and their relation to so-called hidden variable fractal functions. These fractal functions are the projections of a continuous function F of which graph is the attractor of a hyperbolic IFS. The class of hidden variable fractal functions is more diverse than F(X, Y), as their values depend continuously on all the hidden variables determining F, thus making them more appealing as interpolants.