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

نویسندگان

  • Heping Wang
  • Sofiya Ostrovska
چکیده

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 qBernstein polynomials in the case of q > 1. The aim of this paper is to present norm estimates in C[0, 1] for the qBernstein basic polynomials and the q-Bernstein operator Bn,q in the case q > 1. While for 0 < q ≤ 1, ‖Bn,q‖ = 1 for all n ∈ N, in the case q > 1, the norm ‖Bn,q‖ increases rather rapidly as n → ∞. We prove here that ‖Bn,q‖ ∼ Cqqn(n−1)/2/n, n → ∞ with Cq = 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.

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عنوان ژورنال:
  • Math. Comput.

دوره 79  شماره 

صفحات  -

تاریخ انتشار 2010