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Article Citation - WoS: 8Citation - Scopus: 10The Norm Estimates for The q-bernstein Operator in The Case q > 1(Amer Mathematical Soc, 2010) Wang, Heping; Ostrovska, SofiyaThe q-Bernstein basis with 0 < q < 1 emerges as an extension of the Bernstein basis corresponding to a stochastic process generalizing Bernoulli trials forming a totally positive system on [0, 1]. In the case q > 1, the behavior of the q-Bernstein basic polynomials on [0, 1] combines the fast increase in magnitude with sign oscillations. This seriously complicates the study of q-Bernstein polynomials in the case of q > 1. The aim of this paper is to present norm estimates in C[0, 1] for the q-Bernstein basic polynomials and the q-Bernstein operator B-n,B-q in the case q > 1. While for 0 < q <= 1, parallel to B-n,B-q parallel to = 1 for all n is an element of N, in the case q > 1, the norm parallel to B-n,B-q parallel to increases rather rapidly as n -> infinity. We prove here that parallel to B-n,B-q parallel to similar to C(q)q(n(n-1)/2)/n, n -> infinity with C-q = 2 (q(-2); q(-2))(infinity)/e. Such a fast growth of norms provides an explanation for the unpredictable behavior of q-Bernstein polynomials (q > 1) with respect to convergence.Article On the Continuity in q of the Family of the Limit q-durrmeyer Operators(de Gruyter Poland Sp Z O O, 2024) Yilmaz, Ovgu Gurel; Ostrovska, Sofiya; Turan, MehmetThis study deals with the one-parameter family {D-q}(q is an element of[0,1]) of Bernstein-type operators introduced by Gupta and called the limit q-Durrmeyer operators. The continuity of this family with respect to the parameter q is examined in two most important topologies of the operator theory, namely, the strong and uniform operator topologies. It is proved that {D-q}(q is an element of[0,1]) is continuous in the strong operator topology for all q is an element of [0, 1]. When it comes to the uniform operator topology, the continuity is preserved solely at q = 0 and fails at all q is an element of (0, 1]. In addition, a few estimates for the distance between two limit q-Durrmeyer operators have been derived in the operator norm on C[0, 1].
