4 results
Search Results
Now showing 1 - 4 of 4
Article Citation - WoS: 36Citation - Scopus: 40On the Improvement of Analytic Properties Under the Limit Q-Bernstein Operator(Academic Press inc Elsevier Science, 2006) Ostrovska, SLet B-n(f, q; x), n = 1, 2,... be the q-Bernstein polynomials of a function f is an element of C[0, 1]. In the case 0 < q < 1, a sequence {B-n(f, q; x)} generates a positive linear operator B-infinity = B-infinity,B-q on C[0, 1], which is called the limit q-Bernstein operator In this paper, a connection between the smoothness of a function f and the analytic properties of its image under Boo is studied. (c) 2005 Elsevier Inc. All rights reserved.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.Article Citation - WoS: 1Citation - Scopus: 1Uncorrelatedness Sets of Bounded Random Variables(Academic Press inc Elsevier Science, 2004) Ostrovska, SAn uncorrelatedness set of two random variables shows which powers of random variables are uncorrelated. These sets provide a measure of independence: the wider an uncorrelatedness set is, the more independent random variables are. Conditions for a subset of N-2 to be an uncorrelatedness set of bounded random variables are studied. Applications to the theory of copulas are given. (C) 2004 Elsevier Inc. All rights reserved.
