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Review Citation - WoS: 5Citation - Scopus: 5A Survey of Results on the Limit q-bernstein Operator(Hindawi Ltd, 2013) Ostrovska, SofiyaThe limit q-Bernstein operator B-q emerges naturally as a modification of the Szasz-Mirakyan operator related to the Euler distribution, which is used in the q-boson theory to describe the energy distribution in a q-analogue of the coherent state. At the same time, this operator bears a significant role in the approximation theory as an exemplary model for the study of the convergence of the q-operators. Over the past years, the limit q-Bernstein operator has been studied widely from different perspectives. It has been shown that. is a positive shape-preserving linear operator on C[0, 1] with parallel to B-q parallel to = 1. Its approximation properties, probabilistic interpretation, the behavior of iterates, and the impact on the smoothness of a function have already been examined. In this paper, we present a review of the results on the limit q-Bernstein operator related to the approximation theory. A complete bibliography is supplied.Article An Elaboration of the Cai-Xu Result on (p, q)-integers(Springer Heidelberg, 2020) Ostrovska, SofiyaThe investigation of the (p, q)-Bernstein operators put forth the problem of finding the conditions when a sequence of (p, q)-integers tends to infinity. This is crucial for justifying the convergence results pertaining to the (p, q)-operators. Recently, Cai and Xu found a necessary and sufficient condition on sequences {p(n)} and {q(n)}, where 0 < q(n) < p(n) <= 1, to guarantee that a sequence of (p(n), q(n))-integers tends to infinity. This article presents an elaborated version of their result.Article Citation - WoS: 1Citation - Scopus: 2Functions Whose Smoothness Is Not Improved Under the Limit q-bernstein Operator(Springer, 2012) Ostrovska, SofiyaThe limit q-Bernstein operator B-q emerges naturally as a modification of the Szasz-Mirakyan operator related to the Euler probability distribution. At the same time, this operator serves as the limit for a sequence of the q-Bernstein polynomials with 0 < q < 1. Over the past years, the limit q-Bernstein operator has been studied widely from different perspectives. Its approximation, spectral, and functional-analytic properties, probabilistic interpretation, the behavior of iterates, and the impact on the analytic characteristics of functions have been examined. It has been proved that under a certain regularity condition, B-q improves the smoothness of a function which does not satisfy the Holder condition. The purpose of this paper is to exhibit 'exceptional' functions whose smoothness is not improved under the limit q-Bernstein operator. MSC: 26A15; 26A16; 41A36Article Citation - WoS: 11Citation - Scopus: 13The Convergence of q-bernstein Polynomials (0 < q < 1) in the Complex Plane(Wiley-v C H verlag Gmbh, 2009) Ostrovska, SofiyaThe paper focuses at the estimates for the rate of convergence of the q-Bernstein polynomials (0 < q < 1) in the complex plane. In particular, a generalization of previously known results on the possibility of analytic continuation of the limit function and an elaboration of the theorem by Wang and Meng is presented. (C) 2009 WILEY-VCH Verlag GmbH & Co. KGaA, WeinheimArticle 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 ON THE IMAGES OF ENTIRE FUNCTIONS UNDER THE LIMIT q-BERNSTEIN OPERATOR(indian Nat Sci Acad, 2017) Ostrovska, SofiyaThe limit q-Bernstein operator B-q comes out naturally as the limit for the sequence of q-Bernstein operators in the case 0 < q < 1 : Alternatively, it can be viewed as a modification of the Szasz-Mirakyan operator related to the Euler distribution. In this paper, a necessary and sufficient condition for a function g to be an image of an entire function under B-q is presented.Article Citation - WoS: 7Citation - Scopus: 7The Approximation of Power Function by the q-bernstein Polynomials in the Case q > 1(Element, 2008) Ostrovska, SofiyaSince for q > 1. q-Bernstein polynomials are not positive linear operators on C[0, 1] the investigation of their convergence properties turns out to be much more difficult than that in the case 0 < q < 1. It is known that, in the case q > 1. the q-Bernstein polynomials approximate the entire functions and, in particular, polynomials uniformly on any compact set in C. In this paper. the possibility of the approximation for the function (z + a)(alpha), a >= 0. with a non-integer alpha > -1 is studied. It is proved that for a > 0, the function is uniformly approximated on any compact set in {z : vertical bar z vertical bar < a}, while on any Jordan arc in {z : vertical bar z vertical bar > a}. the uniform approximation is impossible, In the case a = 0(1) the results of the paper reveal the following interesting phenomenon: the power function z(alpha), alpha > 0: is approximated by its, q-Bernstein polynomials either on any (when alpha is an element of N) or no (when alpha is not an element of N) Jordan arc in C.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: 2Citation - Scopus: 2On the Properties of the Limit q-bernstein Operator(Akademiai Kiado Zrt, 2011) Ostrovska, SofiyaThe limit q-Bernstein operator B-q = B-infinity,B-q : C [0, 1]. C [0, 1] emerges naturally as a q-version of the Szasz-Mirakyan operator related to the q-deformed Poisson 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 B-q is a positive shape-preserving linear operator on Cinverted right perpendicular0, 1inverted left perpendicular with. parallel to B-q parallel to = 1. Its approximation properties, probabilistic interpretation, behavior of iterates, and the impact on the smoothness have been examined. In this paper, it is shown that the possibility of an analytic continuation of B(q)f into {z : vertical bar z vertical bar < R}, R > 1, implies the smoothness of f at 1, which is stronger when R is greater. If B(q)f can be extended to an entire function, then f is infinitely differentiable at 1, and a sufficiently slow growth of B(q)f implies analyticity of f in {z : vertical bar z-1 vertical bar < delta}, where delta is greater when the growth is slower. Finally, there is a bound for the growth of B(q)f which implies f to be an entire function.Article Citation - Scopus: 2The Approximation of Power Function by the Q-Bernstein Polynomials in the Case Q > 1(Element D.O.O., 2008) Ostrovska,S.Since for q > 1, q-Bernstein polynomials are not positive linear operators on C[0, 1], the investigation of their convergence properties turns out to be much more difficult than that in the case 0 < q < 1. It is known that, in the case q > 1, the q-Bernstein polynomials approximate the entire functions and, in particular, polynomials uniformly on any compact set in ℂ. In this paper, the possibility of the approximation for the function (z + a)α, a ≥ 0, with a non-integer α > -1 is studied. It is proved that for a > 0, the function is uniformly approximated on any compact set in {z: \z| < a}, while on any Jordan arc in {z: \z\ > a}, the uniform approximation is impossible. In the case a = 0, the results of the paper reveal the following interesting phenomenon: the power function zα, α > 0, is approximated by its q-Bernstein polynomials either on any (when α ∈ ℕ) or no (when α ∉ ℕ) Jordan arc in ℂ.
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