On the Convergence and Iterates of q-Bernstein Polynomials

Abstract

The convergence properties of q-Bernstein polynomials are investigated. When q⩾1 is fixed the generalized Bernstein polynomials B n f of f, a one parameter family of Bernstein polynomials, converge to f as n→∞ if f is a polynomial. It is proved that, if the parameter 0< q<1 is fixed, then B n f→ f if and only if f is linear. The iterates of B n f are also considered. It is shown that B n M f converges to the linear interpolating polynomial for f at the endpoints of [0,1], for any fixed q>0, as the number of iterates M→∞. Moreover, the iterates of the Boolean sum of B n f converge to the interpolating polynomial for f at n+1 geometrically spaced nodes on [0,1].