Non-Asymptotic Norm Estimates for the Q-Bernstein Operators

Abstract

The aim of this paper is to present new non-asymptotic norm estimates in C[0,1] for the q-Bernstein operators Bn,q in the case q > 1. While for 0 < q ≤ 1, {double pipe}Bn,q{double pipe} = 1 for all n ∈ ℕ, in the case q > 1, the norm {double pipe}Bn,q{double pipe} grows rather rapidly as n → + ∞ and q → + ∞. Both theoretical and numerical comparisons of the new estimates with the previously available ones are carried out. The conditions are determined under which the new estimates are better than the known ones. © Springer Science+Business Media New York 2013.