BERNSTEIN FRACTAL RATIONAL APPROXIMANTS WITH NO CONDITION ON SCALING VECTORS

Abstract

Fractal functions defined through iterated function system have been successfully used to approximate any continuous real-valued function defined on a compact interval. The fractal dimension is a quantifier (or index) of irregularity (non-differentiability) of fractal approximant and it depends on the scaling factors of the fractal approximant. Viswanathan and Chand [Approx. Theory 185 (2014) 31–50] studied fractal rational approximation under the hypothesis “magnitude of the scaling factors goes to zero”. In this paper, first, we introduce a new class of fractal approximants, namely, Bernstein [Formula: see text]-fractal functions which converge to the original function for every scaling vector. Using the proposed class of fractal approximants and imposing no condition on the corresponding scaling factors, we establish self-referential Bernstein [Formula: see text]-fractal rational functions and their approximation properties. In particular, (i) we study the fractal analogue of the Weierstrass theorem and Müntz theorem of rational functions, (ii) we study the one-sided approximation by Bernstein [Formula: see text]-fractal rational functions, (iii) we develop copositive Bernstein fractal rational approximation, (iv) we investigate the existence of a minimizing sequence of fractal rational approximation to a continuous function defined as a real compact interval. Finally, we introduce the non-self-referential Bernstein [Formula: see text]-fractal approximants.