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Article Citation - WoS: 16Citation - Scopus: 16q-bernstein Polynomials of the Cauchy Kernel(Elsevier Science inc, 2008) Ostrovska, SofiyaDue to the fact that in the case q > 1, q-Bernstein polynomials are not positive linear operators on C[0, 1], the study of their approximation properties is essentially more difficult than that for 0 < q < 1. Despite the intensive research conducted in the area lately, the problem of describing the class of functions in C[0, 1] uniformly approximated by their q-Bernstein polynomials (q > 1) is still open. In this paper, the q-Bernstein polynomials B-n,B-q(f(a); z) of the Cauchy kernel f(a) = 1/(z - a), a is an element of C \ [0, 1] are found explicitly and their properties are investigated. In particular, it is proved that if q > 1, then polynomials B-n,B-q(f(a); z) converge to f(a) uniformly on any compact set K subset of {z : vertical bar z vertical bar < vertical bar a vertical bar}. This result is sharp in the following sense: on any set with an accumulation point in {z : vertical bar z vertical bar > vertical bar a vertical bar}, the sequence {B-n,B-q(f(a); z) is not even uniformly bounded. (C) 2007 Elsevier Inc. All rights reserved.Article Citation - WoS: 11Citation - Scopus: 15The q-versions of the Bernstein Operator: From Mere Analogies To Further Developments(Springer Basel Ag, 2016) Ostrovska, SofiyaThe article exhibits a review of results on two popular q-versions of the Bernstein polynomials, namely, the LupaAY q-analogue and the q-Bernstein polynomials. Their similarities and distinctions are discussed.Article Citation - WoS: 15Citation - Scopus: 18The Sharpness of Convergence Results for q-bernstein Polynomials in The Case q > 1(Springer Heidelberg, 2008) Ostrovska, SofiyaDue to the fact that in the case q > 1 the q-Bernstein polynomials are no longer positive linear operators on C[0, 1], the study of their convergence properties turns out to be essentially more difficult than that for q 1. In this paper, new saturation theorems related to the convergence of q-Bernstein polynomials in the case q > 1 are proved.Article Citation - WoS: 8Citation - Scopus: 10On the Image of the Limit q-bernstein Operator(Wiley, 2009) Ostrovska, SofiyaThe limit q-Bernstein operator B-q emerges naturally as an analogue to the Szasz-Mirakyan operator related to the Euler distribution. Alternatively, B-q comes out as a limit for a sequence of q-Bernstein polynomials in the case 0Article Citation - WoS: 3Citation - Scopus: 4The Unicity Theorems for the Limit Q-Bernstein Operator(Taylor & Francis Ltd, 2009) Ostrovska, SofiyaThe limit q-Bernstein operator [image omitted] emerges naturally as a q-version of the Szasz-Mirakyan operator related to the Euler distribution. The latter is used in the q-boson theory to describe the energy distribution in a q-analogue of the coherent state. The limit q-Bernstein operator has been widely studied lately. It has been shown that [image omitted] is a positive shape-preserving linear operator on [image omitted] with [image omitted] Its approximation properties, probabilistic interpretation, the behaviour of iterates, eigenstructure and the impact on the smoothness of a function have been examined. In this article, we prove the following unicity theorem for operator: if f is analytic on [0, 1] and [image omitted] for [image omitted] then f is a linear function. The result is sharp in the following sense: for any proper closed subset [image omitted] of [0, 1] satisfying [image omitted] there exists a non-linear infinitely differentiable function f so that [image omitted] for all [image omitted].
