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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: 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 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: 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 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: 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].
