THE NORM ESTIMATES FOR THE q-BERNSTEIN OPERATOR

The q-Bernstein basis with 0 < q < 1 emerges as an extension of the Bernstein basis corresponding to a stochastic process generalizing Bernoulli trials forming a totally positive system on [0, 1]. In the case q > 1, the behavior of the q-Bernstein basic polynomials on [0, 1] combines the fast increase in magnitude with sign oscillations. This seriously complicates the study of qBernstein polynomials in the case of q > 1. The aim of this paper is to present norm estimates in C[0, 1] for the qBernstein basic polynomials and the q-Bernstein operator Bn,q in the case q > 1. While for 0 < q ≤ 1, ‖Bn,q‖ = 1 for all n ∈ N, in the case q > 1, the norm ‖Bn,q‖ increases rather rapidly as n → ∞. We prove here that ‖Bn,q‖ ∼ Cqqn(n−1)/2/n, n → ∞ with Cq = 2 (q−2; q−2)∞/e. Such a fast growth of norms provides an explanation for the unpredictable behavior of q-Bernstein polynomials (q > 1) with respect to convergence.