# A fractalization of rational trigonometric function

## TL;DR

This paper introduces a new class of self-referential fractal rational trigonometric functions, establishes approximation theorems, and corrects previous errors in related fractal function literature.

## Contribution

It defines fractal rational trigonometric functions, proves approximation theorems, and addresses errors in recent studies on fractal functions.

## Key findings

- Existence of best fractal rational trigonometric approximants
- Upper bounds for approximation errors
- Correction of mathematical errors in recent fractal function literature

## Abstract

In [14,26], 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 [26] and followed in the setting of Bernstein fractal rational functions in [23]. 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 [22-24].