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Article 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: 15Citation - Scopus: 14Generalized Transportation Cost Spaces(Springer Basel Ag, 2019) Ostrovska, Sofiya; Ostrovskii, Mikhail I.The paper is devoted to the geometry of transportation cost spaces and their generalizations introduced by Melleray et al. (Fundam Math 199(2):177-194, 2008). Transportation cost spaces are also known as Arens-Eells, Lipschitz-free, or Wasserstein 1 spaces. In this work, the existence of metric spaces with the following properties is proved: (1) uniformly discrete metric spaces such that transportation cost spaces on them do not contain isometric copies of l(1), this result answers a question raised by Cuth and Johanis (Proc Am Math Soc 145(8):3409-3421, 2017); (2) locally finite metric spaces which admit isometric embeddings only into Banach spaces containing isometric copies of l(1); (3) metric spaces for which the double-point norm is not a norm. In addition, it is proved that the double-point norm spaces corresponding to trees are close to l(infinity)(d) of the corresponding dimension, and that for all finite metric spaces M, except a very special class, the infimum of all seminorms for which the embedding of M into the corresponding seminormed space is isometric, is not a seminorm.Article Citation - WoS: 11Citation - Scopus: 15The q-versions of the Bernstein Operator: From Mere Analogies To Further Developments(Springer Basel Ag, 2016) Ostrovska, SofiyaThe article exhibits a review of results on two popular q-versions of the Bernstein polynomials, namely, the LupaAY q-analogue and the q-Bernstein polynomials. Their similarities and distinctions are discussed.Article Citation - WoS: 2Citation - Scopus: 2On the eigenfunctions of the q-Bernstein operators(Springer Basel Ag, 2023) Ostrovska, Sofiya; Turan, MehmetThe eigenvalue problems for linear operators emerge in various practical applications in physics and engineering. This paper deals with the eigenvalue problems for the q-Bernstein operators, which play an important role in the q-boson theory of modern theoretical physics. The eigenstructure of the classical Bernstein operators was investigated in detail by S. Cooper and S. Waldron back in 2000. Some of their results were extended for other Bernstein-type operators, including the q-Bernstein and the limit q-Bernstein operators. The current study is a pursuit of this research. The aim of the present work is twofold. First, to derive for the q-Bernstein polynomials analogues of the Cooper-Waldron results on zeroes of the eigenfunctions. Next, to present in detail the proof concerning the existence of non-polynomial eigenfunctions for the limit q-Bernstein operator.Article On the Convergence of the q-bernstein Polynomials for Power Functions(Springer Basel Ag, 2021) Ostrovska, Sofiya; Ozban, Ahmet YasarThe aim of this paper is to present new results related to the convergence of the sequence of the complex q-Bernstein polynomials {B-n,B-q(f(alpha); z)}, where 0 < q not equal 1 and f(alpha) = x(alpha), alpha >= 0, is a power function on [0, 1]. This study makes it possible to describe all feasible sets of convergence K for such polynomials. Specifically, if either 0 < q < 1 or alpha is an element of N-0, then K = C, otherwise K = {0} boolean OR {q(-j)}(j=0)(infinity). In the latter case, this identifies the sequence K = {0} boolean OR {q(-j)}(j=0)(infinity) as the 'minimal' set of convergence for polynomials B-n,B-q(f; z), f is an element of C[0, 1] in the case q > 1. In addition, the asymptotic behavior of the polynomials {B-n,B-q(f(alpha); z)}, with q > 1 has been investigated and the obtained results are illustrated by numerical examples.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: 3Citation - Scopus: 2The Approximation of All Continuous Functions on [0,1] by q-bernstein Polynomials in the Case q → 1+(Springer Basel Ag, 2008) Ostrovska, SofiyaSince for q > 1, the q-Bernstein polynomials B-n,B-q(f;.) are not positive linear operators on C[0, 1], their convergence properties are not similar to those in the case 0 < q = 1. It has been known that, in general, B-n,B-qn(f;.) does not approximate f is an element of C[0, 1] if q(n) -> 1(+), n ->infinity, unlike in the case q(n) -> 1(-). In this paper, it is shown that if 0 <= q(n) - 1 = o(n(-1)3(-n)), n -> infinity, then for any f is an element of C[0, 1], we have: B-n,B-qn(f; x) -> f(x) as n -> infinity, uniformly on [ 0,1].Article Moment Determinacy Versus q-moment Determinacy of Probability Distributions(Springer Basel Ag, 2021) Ostrovska, Sofiya; Turan, MehmetSince the classical moment problem is an important issue deeply connected to various mathematical disciplines, its q-analogue based on the notion of q-moments has emerged in the study of q-distributions. For a wide class of probability distributions, both of these problems can be considered. The aim of this work is to establish a connection between the two moment problems. In this paper, the class A of probability distributions possessing finite moments of all orders and support on (0, infinity) is examined. For each q is an element of(0,1), a distribution P is an element of A can be characterized with respect to moment-determinacy as well as q-moment determinacy. It is proved that the properties of P regarding these characterizations may differ, and that the q-moment determinacy of P may depend on the value of q.
