THE NORM ESTIMATES FOR THE <i>q</i>-BERNSTEIN OPERATOR IN THE CASE <i>q</i> &gt; 1

Abstract

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 q-Bernstein polynomials in the case of q > 1. The aim of this paper is to present norm estimates in C[0, 1] for the q-Bernstein basic polynomials and the q-Bernstein operator B-n,B-q in the case q > 1. While for 0 < q <= 1, parallel to B-n,B-q parallel to = 1 for all n is an element of N, in the case q > 1, the norm parallel to B-n,B-q parallel to increases rather rapidly as n -> infinity. We prove here that parallel to B-n,B-q parallel to similar to C(q)q(n(n-1)/2)/n, n -> infinity with C-q = 2 (q(-2); q(-2))(infinity)/e. Such a fast growth of norms provides an explanation for the unpredictable behavior of q-Bernstein polynomials (q > 1) with respect to convergence.