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Article Citation - WoS: 71Citation - Scopus: 74On 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: 1Citation - Scopus: 1The Distance Between Two Limit q-bernstein Operators(Rocky Mt Math Consortium, 2020) Ostrovska, Sofiya; Turan, MehmetFor q is an element of (0, 1), let B-q denote the limit q-Bernstein operator. The distance between B-q and B-r for distinct q and r in the operator norm on C[0, 1] is estimated, and it is proved that 1 <= parallel to B-q - B-r parallel to <= 2, where both of the equalities can be attained. Furthermore, the distance depends on whether or not r and q are rational powers of each other. For example, if r(j) not equal q(m) for all j, m is an element of N, then parallel to B-q - B-r parallel to = 2, and if r = q(m) for some m is an element of N, then parallel to B-q - B-r parallel to = 2(m - 1)/m.Article On the Oscillation of Discrete Volterra Equations With Positive and Negative Nonlinearities(Rocky Mt Math Consortium, 2018) Ozbekler, AbdullahIn this paper, we give new oscillation criteria for discrete Volterra equations having different nonlinearities such as super-linear and sub-linear cases. We also present some new sufficient conditions for oscillation under the effect of the oscillatory forcing term.Article Citation - WoS: 1Citation - Scopus: 1The Convergence of q-bernstein Polynomials (0 < q < 1) and Limit q-bernstein Operators in Complex Domains(Rocky Mt Math Consortium, 2009) Ostrovska, Sofiya; Wang, HepingDue to the fact that the convergence properties of q-Bernstein polynomials are not similar to those in the classical case q = 1, their study has become an area of intensive research with a wide scope of open problems and unexpected results. The present paper is focused on the convergence of q-Bernstein polynomials, 0 < q < 1, and related linear operators in complex domains. An analogue of the classical result on the simultaneous approximation is presented. The approximation of analytic functions With the help of the limit q-Bernstein operator is studied.
