Bernstein fractal approximation and fractal full Müntz theorems

Abstract

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 $\\alpha$-fractal functions, which converge to the given continuous functioneven if the magnitude of the scaling factors does not approach zero.We use Bernstein $\\alpha$-fractal functions toconstruct the sequence of Bernstein Müntz fractal polynomials that converges toeither $f\\in \\mathcal{C}(I)$ or $f\\in L^p(I), 1 \\le p < \\infty.$ 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 $f\\in \\mathcal{C}(I),$ we developa double sequence $\\big\\{\\{f_{n,l}^{\\alpha}(x)\\}^\\infty_{l=1}\\big\\}^\\infty_{n=1}$ of Bernstein $\\alpha$-fractal functionsthat converges uniformly to $f$. By establishing suitable conditions on thescaling factors, we solve a constrained approximation problem of Bernstein $\\alpha$-fractal Müntz polynomials.We also study the convergence of Bernstein fractal Chebyshev series.