Shape-preserving properties of [formula omitted]-Bernstein polynomials

Abstract

In this paper, we discuss shape-preserving properties of the ω,q-Bernstein polynomials Bnω,q(f;x) introduced by Lewanowicz and Wozny in [S. Lewanowicz, P. Woźny, Generalized Bernstein polynomials, BIT 44(1) (2004) 63–78] for ω,q∈(0,1). When ω=0, we recover the q-Bernstein polynomials introduced by Phillips [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. For ω,q∈(0,1), we show that the basic ω,q-Bernstein polynomial basis is a normalized totally positive basis on [0,1] and that the ω,q-Bernstein operators Bnω,q on C[0,1] are variation-diminishing, monotonicity-preserving and convexity-preserving. We also show that the ω,q-Bernstein polynomials of a convex function f in the case ω,q∈(0,1) are monotonic in the parameters ω and q, for fixed n..