A Fractalization of Rational Trigonometric Functions

Abstract

In Navascues (Z Anal Anwend 25(2):401–418, 2005) and Viswanathan and Chand (J Approx Theory 185:31–50, 2014), new approximation classes of self-referential functions are introduced as fractal versions of the classes of polynomials and rational functions. As a sequel, in the present article, we define a new approximation class consisting of self-referential functions, referred to as the fractal rational trigonometric functions. We establish Weierstrass type approximation theorems for this class and prove the existence of a best fractal rational trigonometric approximant to a real-valued continuous function on a compact interval. Furthermore, we provide an upper bound for the smallest error in approximating a prescribed continuous function by a fractal rational trigonometric function. This extemporizes an analogous result in the context of fractal rational function appeared in Viswanathan and Chand (J Approx Theory 185:31–50, 2014) and followed in the setting of Bernstein fractal rational functions in Vijender (Fractals 26(4):1850045, 2018). The last part of the article aims to clarify and correct the mathematical errors in some results on the Bernstein $$\alpha $$-fractal functions appeared recently in the literature (Vijender in Acta Appl Math 159:11–27, 2019, Vijender in Fractals 26(4):1850045, 2018, Vijender in Mediterr J Math 211:1–16, 2018).