Bernstein fractal approximation and fractal full Müntz theorems

摘要

Fractal interpolation functions defined by means of suitable Iterated Function Systemsprovide 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, namelyBernstein $lpha$-fractal functions, which converge to the given continuous functioneven if the magnitude of the scaling factors does not approach zero.We use Bernstein $lpha$-fractal functions toconstruct the sequence of Bernstein Müntz fractal polynomials that converges toeither $fin mathcal{C}(I)$ or $fin L^p(I), 1 le p infty.$ This gives afractal analogue of the full Müntz theorems in the aforementioned function spaces.For a given sequence ${f_n(x)}^{infty}_{n=1}$ of continuous functionsthat converges uniformly to a function $fin mathcal{C}(I),$ we developa double sequence $ig{{f_{n,l}^{lpha}(x)}^infty_{l=1}ig}^infty_{n=1}$ of Bernstein $lpha$-fractal functionsthat converges uniformly to $f$. By establishing suitable conditions on thescaling factors, we solve a constrained approximation problem of Bernstein $lpha$-fractal Müntz polynomials.We also study the convergence of Bernstein fractal Chebyshev series.