Bernstein Fractal Trigonometric Approximation

Abstract

Fractal interpolation and approximation received a lot of attention in the last thirty years. The main aim of the current article is to study a fractal trigonometric approximants which converge to the given continuous function even if the magnitude of the scaling factors does not approach zero. In this paper, we first introduce a new class of fractal approximants, namely, Bernstein $\alpha$-fractal functions using the theory of fractal approximation and Bernstein polynomial. Using the proposed class of fractal approximants and imposing no condition on corresponding scaling factors, we establish that the set of Bernstein $\alpha$-fractal trigonometric functions is fundamental in the space of continuous periodic functions. Fractal version of Gauss formula of trigonometric interpolation is obtained by means of Bernstein trigonometric fractal polynomials. We study the Bernstein fractal Fourier series of a continuous periodic function $f$ defined on $[-l,l]$. The Bernstein fractal Fourier series converges to $f$ even if the magnitude of the scaling factors does not approach zero. Existence of the $\mathcal{C}^{r}$-Bernstein fractal functions is investigated, and Bernstein cubic spline fractal interpolation functions are proposed based on the theory of $\mathcal{C}^{r}$-Bernstein fractal functions.