Volume 51, pp. 1-14, 2019.

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 α-fractal functions, which converge to the given continuous function even if the magnitude of the scaling factors does not approach zero. We use Bernstein α-fractal functions to construct the sequence of Bernstein Müntz fractal polynomials that converges to either fC(I) or fLp(I),1p<. This gives a fractal analogue of the full Müntz theorems in the aforementioned function spaces. For a given sequence {fn(x)}n=1 of continuous functions that converges uniformly to a function fC(I), we develop a double sequence {{fn,lα(x)}l=1}n=1 of Bernstein α-fractal functions that converges uniformly to f. By establishing suitable conditions on the scaling factors, we solve a constrained approximation problem of Bernstein α-fractal Müntz polynomials. We also study the convergence of Bernstein fractal Chebyshev series.

Full Text (PDF) [503 KB], BibTeX , DOI: 10.1553/etna_vol51s1

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: α-fractal rational splines for constrained interpolation