On the Convergence and Iterates of q-Bernstein Polynomials

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