33 results
Search Results
Now showing 1 - 10 of 33
Article Citation - WoS: 72Citation - Scopus: 75On the Lupas q-analogue of the Bernstein Operator(Rocky Mt Math Consortium, 2006) Ostrovska, SofiyaLet R-n(f,q;x) : C[0, 1] -> C[0, 1] be q-analogues of the Bernstein operators defined by Lupas in 1987. If q = 1, then R-n (f, 1; x) are classical Bernstein polynomials. For q not equal 1, the operators R-n (f, q; x) are rational functions rather than polynomials. The paper deals with convergence properties of the sequence {R-n (f, q; x)}. It is proved that {R-n (f, q(n); x)} converges uniformly to f(x) for any f(x) is an element of C[0, 1] if and only if q(n) -> 1. In the case q > 0, q not equal 1 being fixed the sequence I R. (f, q; x) I converges uniformly to f(x) is an element of C[0, 1] if and only if f(x) is linear.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: 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 Weak Uncorrelatedness of Random Variables(Springer, 2006) Ostrovska, SNew measures of independence for n random variables, based on their moments, are studied. A scale of degrees of independence for random variables which starts with uncorrelatedness (for n = 2) and finishes at independence is constructed. The scale provides a countable linearly ordered set of measures of independence.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: 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: 2On the q-bernstein Polynomials of the Logarithmic Function in the Case q > 1(Walter de Gruyter Gmbh, 2016) Ostrovska, SofiyaThe q-Bernstein basis used to construct the q-Bernstein polynomials is an extension of the Bernstein basis related to the q-binomial probability distribution. This distribution plays a profound role in the q-boson operator calculus. In the case q > 1, q-Bernstein basic polynomials on [0, 1] combine 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 new results related to the q-Bernstein polynomials B-n,B- q of discontinuous functions in the case q > 1. The behavior of polynomials B-n,B- q(f; x) for functions f possessing a logarithmic singularity at 0 has been examined. (C) 2016 Mathematical Institute Slovak Academy of SciencesArticle 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 - 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.
