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The norm estimates of the q-Bernstein operators for varying q>1. (English) Zbl 1236.41026
Summary: The aim of this paper is to present norm estimates in C[0,1] for the q-Bernstein basic polynomials and the q-Bernstein operators B n,q in the case q>1. While for for all n, in the case q>1, the norm B n,q increases rather rapidly as q+. In this study, it is proved that B n,q C n q n(n-1)/2 , q+ with C n =2 n(1-1 n) n-1 . Moreover, it is shown that B n,q 2q n(n-1)/2 ne as n, q. The results of the paper are illustrated by numerical examples.
MSC:
41A35Approximation by operators (in particular, by integral operators)
41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
05A30q-calculus and related topics
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