The norm estimates for the $q$-Bernstein operator in the case $q>1$

Abstract

The q-Bernstein basis with 0 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,q in the case q > 1. While for 0 1, the norm ∥B n,q ∥ increases rather rapidly as n → oo. We prove here that ∥B n,q ∥ ~ C q q n(n-1)/2 /n, n → oo with C q = 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.