q-Bernstein polynomials and their iterates

Let B-n (f,q;x), n = 1,2,... be q-Bernstein polynomials of a function f: [0, 1] --> C. The polynomials B-n(f, 1; x) are classical Bernstein polynomials. For q not equal 1 the properties of q-Bernstein polynomials differ essentially from those in the classical case. This paper deals with approximating properties of q-Bernstein polynomials in the case q>1 with respect to both n and q. Some estimates on the rate of convergence are given. In particular, it is proved that for a function f analytic in {z: \z\ < q + ε} the rate of convergence of {B-n(f, q; x)} to f (x) in the norm of C[0, 1] has the order q(-n) (versus 1/n for the classical Bernstein polynomials). Also iterates of q-Bernstein polynomials {B-n(jn) (f, q; x)}, where both n --> infinity and j(n) --> infinity, are studied. It is shown that for q is an element of (0, 1) the asymptotic behavior of such iterates is quite different from the classical case. In particular, the limit does not depend on the rate of j(n) --> infinity. (C) 2003 Elsevier Science (USA). All rights reserved.