Zipper Fractal Functions with Variable Scalings

Abstract

Zipper fractal interpolation function (ZFIF) is a generalization of fractal interpolation function through an
 improved version of iterated function system by using a binary parameter called a signature. The signature
 allows the horizontal scalings to be negative. ZFIFs have a complex geometric structure, and they can
 be non-differentiable on a dense subset of an interval I. In this paper, we construct k-times continuously
 differentiable ZFIFs with variable scaling functions on I. Some properties like the positivity, monotonicity,
 and convexity of a zipper fractal function and the one-sided approximation for a continuous function by a
 zipper fractal function are studied. The existence of Schauder basis of zipper fractal functions for the space
 of k-times continuously differentiable functions and the space of p-integrable functions for p ∈ [1,∞) are
 studied. We introduce the zipper versions of full Müntz theorem for continuous function and p-integrable
 functions on I for p ∈ [1,∞).