Non-stationary zipper $$\alpha $$-fractal functions and associated fractal operator

Abstract

The present paper aims to introduce a new concept of a non-stationary scheme for the so called fractal functions. Here we work with a sequence of maps for the zipper iterated function systems (IFS). We show that the proposed method generalizes the existing stationary interpolant in the sense of IFS. Further, we study the elementary properties of the proposed interpolant and calculate its box and Hausdorff dimension. Also, we obtain an upper bound of the graph of the fractional integral of the proposed interpolant. We notice that the box dimension of the graph of the proposed interpolant is independent of the signature value for a fixed scale vector. In the end, using the method of fractal perturbation of a given function, we construct the associated fractal operator and study some of its properties.