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Article Citation - WoS: 7Citation - Scopus: 7On the q-bernstein Polynomials of Unbounded Functions With q > 1(Hindawi Ltd, 2013) Ostrovska, Sofiya; Ozban, Ahmet YasarThe aim of this paper is to present new results related to the q-Bernstein polynomials B-n,B-q (f;x) of unbounded functions in the case q > 1 and to illustrate those results using numerical examples. As a model, the behavior of polynomials B-n,B-q (f;x) is examined both theoretically and numerically in detail for functions on [0, 1] satisfying f(x) similar to Kx(-alpha) as x -> 0(+), where alpha > 0 and K not equal 0 are real numbers.Review Citation - WoS: 5Citation - Scopus: 5A Survey of Results on the Limit q-bernstein Operator(Hindawi Ltd, 2013) Ostrovska, SofiyaThe limit q-Bernstein operator B-q emerges naturally as a modification of the Szasz-Mirakyan operator related to the Euler distribution, which is used in the q-boson theory to describe the energy distribution in a q-analogue of the coherent state. At the same time, this operator bears a significant role in the approximation theory as an exemplary model for the study of the convergence of the q-operators. Over the past years, the limit q-Bernstein operator has been studied widely from different perspectives. It has been shown that. is a positive shape-preserving linear operator on C[0, 1] with parallel to B-q parallel to = 1. Its approximation properties, probabilistic interpretation, the behavior of iterates, and the impact on the smoothness of a function have already been examined. In this paper, we present a review of the results on the limit q-Bernstein operator related to the approximation theory. A complete bibliography is supplied.Article Citation - WoS: 1Citation - Scopus: 2The Functional-Analytic Properties of the Limit q-bernstein Operator(Hindawi Ltd, 2012) Ostrovska, SofiyaThe limit q-Bernstein operator B-q, 0 < q < 1, emerges naturally as a modification of the Szasz-Mirakyan operator related to the Euler distribution. The latter is used in the q-boson theory to describe the energy distribution in a q-analogue of the coherent state. Lately, the limit q-Bernstein operator has been widely under scrutiny, and it has been shown that B-q is a positive shape-preserving linear operator on C[0, 1] with parallel to B-q parallel to = 1. Its approximation properties, probabilistic interpretation, eigenstructure, and impact on the smoothness of a function have been examined. In this paper, the functional-analytic properties of B-q are studied. Our main result states that there exists an infinite-dimensional subspace M of C[0, 1] such that the restriction B-q vertical bar(M) is an isomorphic embedding. Also we show that each such subspace M contains an isomorphic copy of the Banach space c(0).
