Fractal bases for Banach spaces of smooth functions

This article explores the properties of fractal interpolation functions with variable scaling parameters, in the context of smooth fractal functions. The first part extends the Barnsley-Harrington theorem for differentiability of fractal functions and the fractal analogue of Hermite interpolation to the present setting. The general result is applied on a special class of iterated function systems in order to develop differentiability of the so-called -fractal functions. This leads to a bounded linear map on the space which is exploited to prove the existence of a Schauder basis for consisting of smooth fractal functions.