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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. Theory185 (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 α -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 α -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 α -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 α -fractal approximants. [ABSTRACT FROM AUTHOR]

One-sided approximation tackles the problem of approximation of a prescribed function by simple traditional functions such as polynomials or trigonometric functions that lie completely above or below it. In this paper, we use the concept of fractal interpolation function (FIF), precisely of fractal trigonometric polynomials, to construct one-sided uniform approximants for some classes of continuous functions. [ABSTRACT FROM AUTHOR]