Properties of convergence for [formula omitted]-Bernstein polynomials

Abstract

In this paper, we discuss properties of the ω , q -Bernstein polynomials B n ω , 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 B n ω , q ( t 2 ; x ) , and demonstrate that if f is convex and ω , q ∈ ( 0 , 1 ) or ( 1 , ∞ ) , then B n ω , q ( f ; x ) are monotonically decreasing in n for all x ∈ [ 0 , 1 ] . We prove that for ω ∈ ( 0 , 1 ) , q n ∈ ( 0 , 1 ] , the sequence { B n ω , q n ( f ) } n ⩾ 1 converges to f uniformly on [ 0 , 1 ] for each f ∈ C [ 0 , 1 ] if and only if lim n → ∞ q n = 1 . For fixed ω , q ∈ ( 0 , 1 ) , we prove that the sequence { B n ω , q ( f ) } converges for each f ∈ C [ 0 , 1 ] and obtain the estimates for the rate of convergence of { B n ω , q ( f ) } by the modulus of continuity of f, and the estimates are sharp in the sense of order for Lipschitz continuous functions.