Properties of convergence for ω,q-Bernstein polynomials☆

In this paper, we discuss properties of the ω,q-Bernstein polynomials Bnω,q(f;x) introduced by S. Lewanowicz and P. Woźny in [S. Lewanowicz, P. Woźny, 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 Bnω,q(t2;x), and demonstrate that if f is convex and ω,q∈(0,1) or (1,∞), then Bnω,q(f;x) are monotonically decreasing in n for all x∈[0,1]. We prove that for ω∈(0,1), qn∈(0,1], the sequence {Bnω,qn(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 {Bnω,q(f)} converges for each f∈C[0,1] and obtain the estimates for the rate of convergence of {Bnω,q(f)} by the modulus of continuity of f, and the estimates are sharp in the sense of order for Lipschitz continuous functions.