The rate of convergence of q-Bernstein polynomials for 0 < q < 1

In the note, we obtain the estimates for the rate of convergence for a sequence of q-Bernstein polynomials [B-n,B-q(f)} for 0 < q < 1 by the modulus of continuity off, and the estimates are sharp with respect to the order for Lipschitz continuous functions. We also get the exact orders of convergence for a family of functions f (x) = x(x), alpha > 0, alpha not equal 1, and the orders do not depend on a, unlike the classical case. (c) 2005 Elsevier Inc. All rights reserved.