Properties of convergence for ω, q-Bernstein polynomials

In this paper, we discuss properties of the omega, q-Bernstein polynomials B-n(omega,q) (f; x) introduced by S. Lewanowicz and P. Wozny in [S. Lewanowicz, P. Wozny, Generalized Bernstein polynomials, BIT 44 (1) (2004) 63-78], where f epsilon C[0, 1], omega, q > 0, omega not equal 1 , q(-1),..., q(-n+1). When omega = 0, we recover the q-Bernstein polynomials introduced by [G.M. Phillips, Bernstein polynomials based on the q-integers, Ann. Numer. Math. 4 (1997) 511-518]; when q = 1, we recover the classical Bernstein polynomials. We compute the second moment of B-n(omega,q) (t(2); x), and demonstrate that if f is convex and omega, q epsilon (0, 1) or (1, infinity), then B-n(omega,q) (f; x) are monotonically decreasing in n for all x epsilon [0, 1]. We prove that for omega epsilon (0, 1), q(n) epsilon (0, 1], the sequence {B-n(omega,qn) (f)} n >= 1 converges to f uniformly on [0, 1] for each f epsilon C [0, 1] if and only if lim(n ->infinity) qn = 1. For fixed omega, q epsilon (0, 1), we prove that the sequence {B-n(omega,q)(f)} converges for each f epsilon C[0, 1] and obtain the estimates for the rate of convergence of (B-n(omega,q) (f)} by the modulus of n n continuity of f, and the estimates are sharp in the sense of order for Lipschitz continuous functions. (c) 2007 Elsevier Inc. All rights reserved.