Properties of convergence for ω, q-Bernstein polynomials ✩

In this paper, we discuss properties of the ω, q-Bernstein polynomials B ω,q n (f ;x) introduced by S. Lewanowicz and P. Wo´zny in [S. Lewanowicz, P. Wo´zny, Generalized Bernstein polynomials, BIT 44 (1) (2004) 63–78], where f ∈ C[0, 1], ω, q > 0, ω = 1, q−1, . . . , q−n+1. When ω = 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 ω,q n (t2;x), and demonstrate that if f is convex and ω, q ∈ (0, 1) or (1,∞), then B ω,q n (f ;x) are monotonically decreasing in n for all x ∈ [0, 1]. We prove that for ω ∈ (0, 1), qn ∈ (0, 1], the sequence {B ω,qn n (f )}n 1 converges to f uniformly on [0, 1] for each f ∈ C[0, 1] if and only if limn→∞qn = 1. For fixed ω, q ∈ (0, 1), we prove that the sequence {B ω,q n (f )} converges for each f ∈ C[0, 1] and obtain the estimates for the rate of convergence of {B ω,q n (f )} by the modulus of continuity of f , and the estimates are sharp in the sense of order for Lipschitz continuous functions. © 2007 Elsevier Inc. All rights reserved.