Generalized zipper fractal approximation and parameter identification problems

This paper introduces a novel technique to approximate a given continuous function f defined on a real compact interval by a new class of zipper alpha-fractal functions which contain a scaling vector and a binary vector or signature. For specific choices of scaling and signature vectors, the corresponding zipper fractal functions simultaneously interpolate and approximate f. When signature is zero, the proposed zipper fractal functions coincide with existing alpha-fractal functions. Hence, the zipper approximation proposed in this manuscript generalizes the existing fractal and classical approximations. Zipper fractal analogue of some elementary results in the classical approximation theory are obtained. Using convex optimization technique, we investigate the existence of optimal zipper fractal function for a given continuous function. The parameter identification problems for zipper alpha-fractal approximants are investigated. We derive sufficient conditions on the parameters of zipper alpha-fractal functions so that these functions preserve the positivity, monotonicity and convexity of the original function f. Also, we have studied the constructions of k-times continuously differentiable zipper alpha-fractal functions and one sided zipper fractal approximants for f. Numerical illustrations are provided to support the proposed theoretical results on zipper alpha-fractal functions.