Properties of convergence for ω,q-Bernstein polynomials

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