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Article 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 - 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.
