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Article Citation - WoS: 9Citation - Scopus: 9The Approximation of Logarithmic Function by q-bernstein Polynomials in the Case q > 1(Springer, 2007) Ostrovska, SofiyaSince 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) remains open. It is known that the approximation occurs for functions admitting an analytic continuation into a disc {z : | z| < R}, R > 1. For functions without such an assumption, no general results on approximation are available. In this paper, it is shown that the function f ( x) = ln( x + a), a > 0, is uniformly approximated by its q-Bernstein polynomials ( q > 1) on the interval [ 0, 1] if and only if a >= 1.Article Citation - WoS: 2Citation - Scopus: 1The Continuity in Q of the Lupaş Q-Analogues of the Bernstein Operators(Academic Press inc Elsevier Science, 2024) Yilmaz, Ovgue Gurel; Turan, Mehmet; Ostrovska, Sofiya; Turan, Mehmet; Ostrovska, Sofiya; Turan, Mehmet; Ostrovska, Sofiya; Mathematics; MathematicsThe Lupas q-analogue Rn,q of the Bernstein operator is the first known q-version of the Bernstein polynomials. It had been proposed by A. Lupas in 1987, but gained the popularity only 20 years later, when q-analogues of classical operators pertinent to the approximation theory became an area of intensive research. In this work, the continuity of operators Rn,q with respect to parameter q in the strong operator topology and in the uniform operator topology has been investigated. The cases when n is fixed and n -> infinity have been considered. (c) 2022 Elsevier Inc. All rights reserved.Article Qualitative results on the convergence of the q-Bernstein polynomials(North Univ Baia Mare, 2015) Ostrovska, Sofiya; Turan, MehmetDespite many common features, the convergence properties of the Bernstein and the q-Bernstein polynomials are not alike. What is more, the cases 0 < q < 1 and q > 1 are not similar to each other in terms of convergence. In this work, new results demonstrating the striking differences which may occur in those convergence properties are presented.Article Citation - WoS: 8Citation - Scopus: 10The Norm Estimates for The q-bernstein Operator in The Case q > 1(Amer Mathematical Soc, 2010) Wang, Heping; Ostrovska, SofiyaThe 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 q-Bernstein polynomials in the case of q > 1. The aim of this paper is to present norm estimates in C[0, 1] for the q-Bernstein basic polynomials and the q-Bernstein operator B-n,B-q in the case q > 1. While for 0 < q <= 1, parallel to B-n,B-q parallel to = 1 for all n is an element of N, in the case q > 1, the norm parallel to B-n,B-q parallel to increases rather rapidly as n -> infinity. We prove here that parallel to B-n,B-q parallel to similar to C(q)q(n(n-1)/2)/n, n -> infinity with C-q = 2 (q(-2); q(-2))(infinity)/e. Such a fast growth of norms provides an explanation for the unpredictable behavior of q-Bernstein polynomials (q > 1) with respect to convergence.Article Citation - WoS: 5Citation - Scopus: 5Analytical Properties of the Lupas q-transform(Academic Press inc Elsevier Science, 2012) Ostrovska, SofiyaThe Lupas q-transform emerges in the study of the limit q-Lupas operator. The latter comes out naturally as a limit for a sequence of the Lupas q-analogues of the Bernstein operator. Given q is an element of (0, 1), f is an element of C left perpendicular0, 1right perpendicular, the q-Lupas transform off is defined by (Lambda(q)f) (z) := 1/(-z; q)(infinity) . Sigma(infinity)(k=0) f(1 - q(k))q(k(k -1)/2)/(q; q)(k)z(k). The transform is closely related to both the q-deformed Poisson probability distribution, which is used widely in the q-boson operator calculus, and to Valiron's method of summation for divergent series. In general, Lambda(q)f is a meromorphic function whose poles are contained in the set J(q) := {-q(-j)}(j=0)(infinity). In this paper, we study the connection between the behaviour of f on leftperpendicular0, 1right perpendicular and the decay of Lambda(q)f as z -> infinity. (C) 2012 Elsevier Inc. All rights reserved.Article Citation - WoS: 3Citation - Scopus: 2The Approximation of All Continuous Functions on [0,1] by q-bernstein Polynomials in the Case q → 1+(Springer Basel Ag, 2008) Ostrovska, SofiyaSince for q > 1, the q-Bernstein polynomials B-n,B-q(f;.) are not positive linear operators on C[0, 1], their convergence properties are not similar to those in the case 0 < q = 1. It has been known that, in general, B-n,B-qn(f;.) does not approximate f is an element of C[0, 1] if q(n) -> 1(+), n ->infinity, unlike in the case q(n) -> 1(-). In this paper, it is shown that if 0 <= q(n) - 1 = o(n(-1)3(-n)), n -> infinity, then for any f is an element of C[0, 1], we have: B-n,B-qn(f; x) -> f(x) as n -> infinity, uniformly on [ 0,1].Article Citation - WoS: 9Citation - Scopus: 8On the Eigenvectors of the q-bernstein Operators(Wiley, 2014) Ostrovska, S.; Turan, M.In this article, both the eigenvectors and the eigenvalues of the q-Bernstein operators have been studied. Explicit formulae are presented for the eigenvectors, whose limit behavior is determined both in the case 01. Because the classical case, where q=1, was investigated exhaustively by S. Cooper and S. Waldron back in 2000, the present article also discusses the related similarities and distinctions with the results in the classical case. Copyright (c) 2013 John Wiley & Sons, Ltd.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.Article Qualitative Results on the Convergence of the Q-Bernstein Polynomials(North University of Baia Mare, 2015) Ostrovska,S.; Turan,M.Despite many common features, the convergence properties of the Bernstein and the q-Bernstein polynomials are not alike. What is more, the cases 0 < q < 1 and q > 1 are not similar to each other in terms of convergence. In this work, new results demonstrating the striking differences which may occur in those convergence properties are presented. © 2015, North University of Baia Mare. All rights reserved.Article Citation - WoS: 8Citation - Scopus: 8On the Lupas q-transform(Pergamon-elsevier Science Ltd, 2011) Ostrovska, SofiyaThe Lupas q-transform emerges in the study of the limit q-Lupas operator. The latter comes out naturally as a limit for a sequence of the Lupas q-analogues of the Bernstein operator. Lately, it has been studied by several authors from different perspectives in mathematical analysis and approximation theory. This operator is closely related to the q-deformed Poisson probability distribution, which is used widely in the q-boson operator calculus. (Lambda(q)f)(z) := 1/(-z; q)(infinity) . Sigma(infinity)(k=0) f(1 - q(k))q(k(k-1)/2)/(q; q)(k) z(k). In this paper, we study some analytic properties of (Lambda(q)f)(z). In particular, we examine the conditions under which Lambda(q)f can either be an entire function, or a rational one. (C) 2010 Elsevier Ltd. All rights reserved.
