Q-Bernstein polynomials and their iterates

Abstract

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≠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 j n ( f,q;x)} , where both n→∞ and j n →∞, are studied. It is shown that for q∈(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 →∞.