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Article Citation - WoS: 3Citation - Scopus: 3The q-bernstein Polynomials of the Cauchy Kernel With a Pole on [0,1] in the Case q > 1(Elsevier Science inc, 2013) Ostrovska, Sofiya; Ozban, Ahmet YasarThe problem to describe the Bernstein polynomials of unbounded functions goes back to Lorentz. The aim of this paper is to investigate the convergence properties of the q-Bernstein polynomials B-n,B-q(f; x) of the Cauchy kernel 1/x-alpha with a pole alpha is an element of [0, 1] for q > 1. The previously obtained results allow one to describe these properties when a pole is different from q(-m) for some m is an element of {0, 1, 2, ...}. In this context, the focus of the paper is on the behavior of polynomials B-n,B-q(f; x) for the functions of the form f(m)(x) = 1/(x - q(-m)), x not equal q(-m) and f(m)(q(-m)) = a, a is an element of R. Here, the problem is examined both theoretically and numerically in detail. (C) 2013 Elsevier Inc. All rights reserved.Article Citation - WoS: 5Citation - Scopus: 6The Norm Estimates of the q-bernstein Operators for Varying q > 1(Pergamon-elsevier Science Ltd, 2011) Ostrovska, Sofiya; Ozban, Ahmet YasarThe 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,B-q in the case q > 1. While for 0 < q <= 1, vertical bar vertical bar B-n,B-q vertical bar vertical bar = 1 for all n is an element of N. in the case q > 1, the norm vertical bar vertical bar B-n,B-q vertical bar vertical bar increases rather rapidly as q -> +infinity. In this study, it is proved that vertical bar vertical bar B-n,B-q vertical bar vertical bar similar to C(n)q(n(n-1)/2), q -> +infinity with C-n = 2/n (1- 1/n)(n-1). Moreover, it is shown that vertical bar vertical bar B-n,B-q vertical bar vertical bar similar to 2q(n(n-1)/2) /ne as n -> infinity, q -> +infinity. The results of the paper are illustrated by numerical examples. (C) 2011 Elsevier Ltd. All rights reserved.
