16 results
Search Results
Now showing 1 - 10 of 16
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: 2Citation - Scopus: 2On the q-bernstein Polynomials of Piecewise Linear Functions in the Case q > 1(Pergamon-elsevier Science Ltd, 2013) Kaskaloglu, Kerem; Ostrovska, SofiyaThe aim of this paper is to present new results related to the approximation of continuous functions by their q-Bernstein polynomials in the case q > 1. The first part of the paper is devoted to the behavior of the q-Bernstein polynomials of piecewise linear functions. This study naturally leads to the notion of truncated q-Bernstein polynomials introduced in the paper. The second part deals with the asymptotic estimates for the norms of the m-truncated q-Bernstein polynomials, in the case where both n and q vary. The results of the paper are illustrated by numerical examples. (C) 2012 Elsevier Ltd. All rights reserved.Article Citation - WoS: 7On the Approximation of Analytic Functions by the q-bernstein Polynomials in the Case q > 1(Kent State University, 2010) Ostrovska, SofiyaSince for q > 1, the q-Bernstein polynomials B(n,q) 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. In this paper, new results on the approximation of continuous functions by the q-Bernstein polynomials in the case q > 1 are presented. It is shown that if f is an element of C[0, 1] and admits an analytic continuation f(z) into {z : |z| < a}, then B(n,q) (f; z) -> f (z) as n -> infinity, uniformly on any compact set in {z : |z| < a}.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 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: 36Citation - Scopus: 40On the Improvement of Analytic Properties Under the Limit Q-Bernstein Operator(Academic Press inc Elsevier Science, 2006) Ostrovska, SLet B-n(f, q; x), n = 1, 2,... be the q-Bernstein polynomials of a function f is an element of C[0, 1]. In the case 0 < q < 1, a sequence {B-n(f, q; x)} generates a positive linear operator B-infinity = B-infinity,B-q on C[0, 1], which is called the limit q-Bernstein operator In this paper, a connection between the smoothness of a function f and the analytic properties of its image under Boo is studied. (c) 2005 Elsevier Inc. All rights reserved.Article Citation - WoS: 2The 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 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: 170Citation - Scopus: 188q-bernstein Polynomials and Their Iterates(Academic Press inc Elsevier Science, 2003) Ostrovska, SLet B-n (f,q;x), n = 1,2,... be q-Bernstein polynomials of a function f: [0, 1] --> C. The polynomials B-n(f, 1; x) are classical Bernstein polynomials. For q not equal 1 the properties of q-Bernstein polynomials differ essentially from those in the classical case. This paper deals with approximating properties of q-Bernstein polynomials in the case q>1 with respect to both n and q. Some estimates on the rate of convergence are given. In particular, it is proved that for a function f analytic in {z: \z\ < q + ε} the rate of convergence of {B-n(f, q; x)} to f (x) in the norm of C[0, 1] has the order q(-n) (versus 1/n for the classical Bernstein polynomials). Also iterates of q-Bernstein polynomials {B-n(jn) (f, q; x)}, where both n --> infinity and j(n) --> infinity, are studied. It is shown that for q is an element of (0, 1) the asymptotic behavior of such iterates is quite different from the classical case. In particular, the limit does not depend on the rate of j(n) --> infinity. (C) 2003 Elsevier Science (USA). All rights reserved.
