On the convergence and iterates of q-Bernstein polynomials

The convergence properties of q-Bernstein polynomials are investigated. When q greater than or equal to 1 is fixed the generalized Bernstein polynomials B(n)f of f, a one parameter family of Bernstein polynomials, converge to f as n --> infinity 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 --> infinity. 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] (C) 2002 Elsevier Science (USA).