## Bernstein fractal approximation and fractal full Müntz theorems

Vijender Nallapu

### Abstract

Fractal interpolation functions defined by means of suitable Iterated Function Systems provide a new framework for the approximation of continuous functions defined on a compact real interval. Convergence is one of the desirable properties of a good approximant. The goal of the present paper is to develop fractal approximants, namely Bernstein $\alpha$-fractal functions, which converge to the given continuous function even if the magnitude of the scaling factors does not approach zero. We use Bernstein $\alpha$-fractal functions to construct the sequence of Bernstein Müntz fractal polynomials that converges to either $f\in \mathcal{C}(I)$ or $f\in L^p(I), 1 \le p < \infty.$ This gives a fractal analogue of the full Müntz theorems in the aforementioned function spaces. For a given sequence $\{f_n(x)\}^{\infty}_{n=1}$ of continuous functions that converges uniformly to a function $f\in \mathcal{C}(I),$ we develop a double sequence $\big\{\{f_{n,l}^{\alpha}(x)\}^\infty_{l=1}\big\}^\infty_{n=1}$ of Bernstein $\alpha$-fractal functions that converges uniformly to $f$. By establishing suitable conditions on the scaling factors, we solve a constrained approximation problem of Bernstein $\alpha$-fractal Müntz polynomials. We also study the convergence of Bernstein fractal Chebyshev series.

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### Key words

Bernstein polynomials, fractal approximation, convergence, full Müntz theorems, Chebyshev series, box dimension.

### AMS subject classifications

41A30, 28A80, 41A17, 41A50.

### Links to the cited ETNA articles

 [16] Vol. 20 (2005), pp. 64-74 M. A. Navascues: Fractal trigonometric approximation [29] Vol. 41 (2014), pp. 420-442 Puthan Veedu Viswanathan and Arya Kumar Bedabrata Chand: $\alpha$-fractal rational splines for constrained interpolation

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