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Article Citation - WoS: 2Citation - Scopus: 2On the q-bernstein Polynomials of the Logarithmic Function in the Case q > 1(Walter de Gruyter Gmbh, 2016) Ostrovska, SofiyaThe q-Bernstein basis used to construct the q-Bernstein polynomials is an extension of the Bernstein basis related to the q-binomial probability distribution. This distribution plays a profound role in the q-boson operator calculus. In the case q > 1, q-Bernstein basic polynomials on [0, 1] combine 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 new results related to the q-Bernstein polynomials B-n,B- q of discontinuous functions in the case q > 1. The behavior of polynomials B-n,B- q(f; x) for functions f possessing a logarithmic singularity at 0 has been examined. (C) 2016 Mathematical Institute Slovak Academy of SciencesArticle On the Lupas q-transform of Unbounded Functions(Walter de Gruyter Gmbh, 2023) Ostrovska, Sofiya; Turan, MehmetThe Lupa , s q-transform comes out naturally in the study of the Lupa , s q-analogue of the Bernstein operator. It is closely related to the Heine q-distribution which has a numerous application in q-boson operator calculus and to the Valiron method of summation for divergent series. In this paper, the Lupa , s q-transform (lambda(q)f)(z), q is an element of (0, 1), of unbounded functions is considered in distinction to the previous researches, where only the case f is an element of C[0, 1] have been investigated. First, the condition for a function to possess the Lupa , s q-transform is presented. Also, results concerning the connection between growth rate of the function f (t) as t -> 1(-) and the growth of its Lupa , s q-transform (lambda(q)f)(z) as z -> infinity are established. (c) 2023 Mathematical Institute Slovak Academy of SciencesArticle The Inversion Results for the Limit q-bernstein Operator(Springer Basel Ag, 2018) Ostrovska, SofiyaThe limit q-Bernstein operator B-q appears as a limit for a sequence of the q-Bernstein or for a sequence of the q-Meyer-Konig and Zeller operators in the case 0 < q < 1. Lately, various features of this operator have been investigated from several angles. It has been proved that the smoothness of f is an element of C[0, 1] affects the possibility for an analytic continuation of its image B-q f. This work aims to investigate the reciprocal: to what extent the smoothness of f can be retrieved from the analytical properties of B-q f.Article Citation - WoS: 8Citation - Scopus: 10On the Image of the Limit q-bernstein Operator(Wiley, 2009) Ostrovska, SofiyaThe limit q-Bernstein operator B-q emerges naturally as an analogue to the Szasz-Mirakyan operator related to the Euler distribution. Alternatively, B-q comes out as a limit for a sequence of q-Bernstein polynomials in the case 0Article Norming Subspaces Isomorphic to l1(Springer Heidelberg, 2015) Ostrovska, SofiyaNorming subspaces are studied widely in the duality theory of Banach spaces. These subspaces are applied to the Borel and Baire classifications of the inverse operators. The main result of this article asserts that the dual of a Banach space X contains a norming subspace isomorphic to l(1) provided that the following two conditions are satisfied: (1) X* contains a subspace isomorphic to l(1); and (2) X* contains a separable norming subspace.Article The Saturation of Convergence for the Complex q-durrmeyer Polynomials(Springer Basel Ag, 2025) Gurel, Ovgu; Ostrovska, Sofiya; Turan, MehmetThe aim of this paper is to establish a saturation result for the complex q-Durrmeyer polynomials (Dn,qf)(z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(D_{n,q}f)(z)$$\end{document}, where q is an element of(0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q \in (0,1)$$\end{document}, f is an element of C[0,1].\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f \in C[0,1].$$\end{document} It is known that the sequence {(Dn,qf)(z)}n is an element of N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{(D_{n,q}f)(z)\}_{n \in {\mathbb {N}}}$$\end{document} converges uniformly on any compact set in C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {C}}$$\end{document} to the limit function (D infinity,qf)(z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(D_{\infty ,q}f)(z)$$\end{document}, which, therefore, is entire. Previously, the rate of this convergence has been estimated as O(qn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(q<^>n)$$\end{document}, n ->infinity.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \rightarrow \infty . $$\end{document} In the present article, this result is refined to derive Voronovskaya-type formula and to demonstrate that this rate is o(qn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$o(q<^>n)$$\end{document}, n ->infinity\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \rightarrow \infty $$\end{document} on a set possessing an accumulation point if and only if f takes on the same value at all qj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q<^>j$$\end{document}, j is an element of N0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j \in {\mathbb {N}}_{0}$$\end{document}.Article Citation - WoS: 2Citation - Scopus: 1The Continuity in Q of the Lupaş Q-Analogues of the Bernstein Operators(Academic Press inc Elsevier Science, 2024) Yilmaz, Ovgue Gurel; Turan, Mehmet; Ostrovska, Sofiya; Turan, Mehmet; Ostrovska, Sofiya; Turan, Mehmet; Ostrovska, Sofiya; Mathematics; MathematicsThe Lupas q-analogue Rn,q of the Bernstein operator is the first known q-version of the Bernstein polynomials. It had been proposed by A. Lupas in 1987, but gained the popularity only 20 years later, when q-analogues of classical operators pertinent to the approximation theory became an area of intensive research. In this work, the continuity of operators Rn,q with respect to parameter q in the strong operator topology and in the uniform operator topology has been investigated. The cases when n is fixed and n -> infinity have been considered. (c) 2022 Elsevier Inc. All rights reserved.Article Citation - WoS: 12Citation - Scopus: 13Assessing Team Work in Engineering Projects(Tempus Publications, 2015) Mishra, Deepti; Ostrovska, Sofiya; Hacaloglu, Tuna; Mathematics; Computer Engineering; Information Systems EngineeringTeam work is considered a valuable teaching technique in higher education. However, the assessment of an individual's work in teams has proved to be a challenging task. Consequently, self-and peer-evaluations are becoming increasingly popular for the assessment of individuals in a team work, though it is essential to determine whether students can judge their own as well as their peer's performance effectively. Self-and peer-evaluations have been applied in different disciplines and their authenticity with regard to teacher's assessment has been evaluated in the literature but this issue has not been investigated in the field of engineering education so far. In this study, a peer-and self-assessment procedure is applied to the evaluation of a project work conducted in teams of 3 or 4 students. The participants were engineering students taking two similar courses related with database design and development. It is found that a majority of the students were unable to assess themselves as objectively as their instructor. Further, it is observed that successful students tend to under-estimate, whereas unsuccessful students tend to over-estimate, their own performance. The paper also establishes that the results of self-assessments are independent from the gender factor.Article Citation - WoS: 11Citation - Scopus: 13The Convergence of q-bernstein Polynomials (0 < q < 1) in the Complex Plane(Wiley-v C H verlag Gmbh, 2009) Ostrovska, SofiyaThe paper focuses at the estimates for the rate of convergence of the q-Bernstein polynomials (0 < q < 1) in the complex plane. In particular, a generalization of previously known results on the possibility of analytic continuation of the limit function and an elaboration of the theorem by Wang and Meng is presented. (C) 2009 WILEY-VCH Verlag GmbH & Co. KGaA, WeinheimArticle Citation - WoS: 16Citation - Scopus: 16q-bernstein Polynomials of the Cauchy Kernel(Elsevier Science inc, 2008) Ostrovska, SofiyaDue to the fact that 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) is still open. In this paper, the q-Bernstein polynomials B-n,B-q(f(a); z) of the Cauchy kernel f(a) = 1/(z - a), a is an element of C \ [0, 1] are found explicitly and their properties are investigated. In particular, it is proved that if q > 1, then polynomials B-n,B-q(f(a); z) converge to f(a) uniformly on any compact set K subset of {z : vertical bar z vertical bar < vertical bar a vertical bar}. This result is sharp in the following sense: on any set with an accumulation point in {z : vertical bar z vertical bar > vertical bar a vertical bar}, the sequence {B-n,B-q(f(a); z) is not even uniformly bounded. (C) 2007 Elsevier Inc. All rights reserved.
