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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: 17Citation - Scopus: 20Positive linear operators generated by analytic functions(Springer, 2007) Ostrovska, SofiyaLet phi be a power series with positive Taylor coefficients {a(k)}(k=0)(infinity) and non-zero radius of convergence r <= infinity. Let xi x, 0 <= x <= r be a random variable whose values alpha(k), k = 0, 1,..., are independent of x and taken with probabilities a(k)x(k)/phi(x), k = 0, 1,.... The positive linear operator (A(phi)f)(x) := E[f(xi x)] is studied. It is proved that if E(xi(x)) = x, E(xi(2)(x)) = qx(2) + bx + c, q, b, c is an element of R, q > 0, then A(phi) reduces to the Szasz-Mirakyan operator in the case q = 1, to the limit q-Bernstein operator in the case 0 < q < 1, and to a modification of the Lupas, operator in the case q > 1.
