Generalised Univariable Fractal Interpolation Functions

Abstract

We show how to construct a generalised iterated function system whose graph is the attractor, a fractal set, of some continuous function which interpolates a given set of data. Moreover, Rakotch contractions and vertical scaling factors as (continuous) ‘contraction functions’ are used in order to obtain generalised fractal interpolation functions with extensive practical applications, including data fitting and approximation of functions. A special generalised fractal interpolation function is introduced as an explicit illustrative example to show the effectiveness of the proposed method as compared to other existing methods. In particular, fractal interpolation functions which are widely presented in the literature can be obtained as particular cases of our construction.