Fractal rational functions and their approximation properties

• The class of fractal rational functions is introduced.
• Approximation properties of this new class is investigated.
• Existence of fractal rational functions copositive with a continuous function is shown.

This article introduces fractal perturbation of rational functions via αα-fractal operator and investigates some approximation theoretic aspects of this new function class, namely, the class of fractal rational functions. Its specific aims are: (i) to define fractal rational functions (ii) to investigate the optimal perturbation to a traditional rational approximant corresponding to a continuous function (iii) to establish the fractal rational function analogues of the celebrated Weierstrass theorem and its generalization, namely, the Müntz theorem (iv) to prove the existence of a best fractal rational approximant to a continuous function defined on a real compact interval, and to study certain properties of the corresponding best approximation operator. By establishing the existence of fractal rational functions that are copositive with a prescribed continuous function, the current article also attempts to invoke fractal functions to the field of shape preserving approximation.