Abstract
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 0 < q ≤ 1 , ‖ B n , q ‖ = 1 for all n ∈ 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 ‖ ∼ 2 q n ( n − 1 ) / 2 n e as n → ∞ , q → + ∞ . The results of the paper are illustrated by numerical examples.
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