Fractal Perturbation of Shaped Functions: Convergence Independent of Scaling

Abstract

In this paper, we introduce a new class of fractal approximants as a fixed points of the Read–Bajraktarevic operator defined on a suitable function space. In the development of our fractal approximants, we used the suitable bounded linear operators defined on the space $${\mathcal {C}}(I)$$ of continuous functions and $$\alpha $$ -fractal functions. The convergence of the proposed fractal approximants towards the continuous function f does not need any condition on the scaling vector. Owing to this reason, the proposed fractal approximants approximate the function f without losing their fractality. We establish constrained approximation by a new class of fractal polynomials. In particular, our constrained fractal polynomials preserve positivity and fractality of the original function simultaneously whenever the original function is positive and irregular. Calculus of the proposed fractal approximants is studied using suitable bounded linear operators defined on the space $${\mathcal {C}}^r(I)$$ of all real-valued functions on the compact interval I that are r-times differentiable with continuous r-th derivative. We identify the IFS parameters so that our $$\alpha $$ -fractal functions preserve fundamental shape properties such as monotonicity and convexity in addition to the smoothness of f in the given compact interval.