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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: 126Citation - Scopus: 136Convergence of Generalized Bernstein Polynomials(Academic Press inc Elsevier Science, 2002) Il'inskii, A; Ostrovska, SLet f is an element of C[0, 1], q is an element of (0, 1), and B-n(f, q; x) be generalized Bernstein polynomials based on the q-integers. These polynomials were introduced by G. M. Phillips in 1997. We study convergence properties of the sequence {B-n(f, q; x)}(n=1)(infinity). It is shown that in general these properties are essentially different from those in the classical case q = 1. (C) 2002 Elsevier Science (USA).Article Citation - WoS: 5Citation - Scopus: 5Analytical Properties of the Lupas q-transform(Academic Press inc Elsevier Science, 2012) Ostrovska, SofiyaThe Lupas q-transform emerges in the study of the limit q-Lupas operator. The latter comes out naturally as a limit for a sequence of the Lupas q-analogues of the Bernstein operator. Given q is an element of (0, 1), f is an element of C left perpendicular0, 1right perpendicular, the q-Lupas transform off is defined by (Lambda(q)f) (z) := 1/(-z; q)(infinity) . Sigma(infinity)(k=0) f(1 - q(k))q(k(k -1)/2)/(q; q)(k)z(k). The transform is closely related to both the q-deformed Poisson probability distribution, which is used widely in the q-boson operator calculus, and to Valiron's method of summation for divergent series. In general, Lambda(q)f is a meromorphic function whose poles are contained in the set J(q) := {-q(-j)}(j=0)(infinity). In this paper, we study the connection between the behaviour of f on leftperpendicular0, 1right perpendicular and the decay of Lambda(q)f as z -> infinity. (C) 2012 Elsevier Inc. All rights reserved.Article Citation - WoS: 170Citation - Scopus: 188q-bernstein Polynomials and Their Iterates(Academic Press inc Elsevier Science, 2003) Ostrovska, SLet B-n (f,q;x), n = 1,2,... be q-Bernstein polynomials of a function f: [0, 1] --> C. The polynomials B-n(f, 1; x) are classical Bernstein polynomials. For q not equal 1 the properties of q-Bernstein polynomials differ essentially from those in the classical case. This paper deals with approximating properties of q-Bernstein polynomials in the case q>1 with respect to both n and q. Some estimates on the rate of convergence are given. In particular, it is proved that for a function f analytic in {z: \z\ < q + ε} the rate of convergence of {B-n(f, q; x)} to f (x) in the norm of C[0, 1] has the order q(-n) (versus 1/n for the classical Bernstein polynomials). Also iterates of q-Bernstein polynomials {B-n(jn) (f, q; x)}, where both n --> infinity and j(n) --> infinity, are studied. It is shown that for q is an element of (0, 1) the asymptotic behavior of such iterates is quite different from the classical case. In particular, the limit does not depend on the rate of j(n) --> infinity. (C) 2003 Elsevier Science (USA). All rights reserved.
