Browsing by Author "Ostrovska, S"
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Article Citation Count: 17The approximation by q-Bernstein polynomials in the case q ↓ 1(Springer Basel Ag, 2006) Ostrovska, Sofiya; MathematicsLet B-n (f, q; x), n = 1, 2, ... , 0 < q < infinity, be the q-Bernstein polynomials of a function f, B-n (f, 1; x) being the classical Bernstein polynomials. It is proved that, in general, {B-n (f, q(n); x)} with q(n) down arrow 1 is not an approximating sequence for f is an element of C[0, 1], in contrast to the standard case q(n) up arrow 1. At the same time, there exists a sequence 0 < delta(n) down arrow 0 such that the condition 1 <= q(n) <= delta(n) implies the approximation of f by {B-n(f, qn; x)} for all f is an element of C[0, 1].Article Citation Count: 127Convergence of generalized Bernstein polynomials(Academic Press inc Elsevier Science, 2002) Ostrovska, Sofiya; Ostrovska, S; MathematicsLet 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 Count: 169q-Bernstein polynomials and their iterates(Academic Press inc Elsevier Science, 2003) Ostrovska, Sofiya; MathematicsLet 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 Count: 38On the improvement of analytic properties under the limit q-Bernstein operator(Academic Press inc Elsevier Science, 2006) Ostrovska, Sofiya; MathematicsLet 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 Count: 1Sets of random variables with a given uncorrelation structure(Elsevier Science Bv, 2001) Ostrovska, Sofiya; MathematicsLet xi (1),...,xi (n) be random variables having finite expectations. Denote i(k) := # {(j(1),...,j(k)): 1 less than or equal to j(1) < ... < j(k) less than or equal to n and E (l=1)pi (k) xi (fi) = (l=1)pi (k) E xi (h)}, k = 2,...,n. The finite sequence (i(2),...,i(n)) is called the uncorrelation structure of xi (1),...,xi (n). It is proved that for any given sequence of nonnegative integers (i(2),...,i(n)) satisfying 0 less than or equal to i(k) less than or equal to ((n)(k))and any given nondegenerate probability distributions P-1,...,P-n there exist random variables eta (1),...,eta (n) with respective distributions P-1,...,P-n such that (i(2),...,i(n)) is their uncorrelation structure. (C) 2001 Elsevier Science B.V. All rights reserved.Article Citation Count: 6Stieltjes classes for M-indeterminate powers of inverse Gaussian distributions(Elsevier Science Bv, 2005) Ostrovska, Sofiya; Stoyanov, J; MathematicsyThe aim of this paper is to exhibit an infinite family (Stieltjes class) of distributions all of which have the same moments as some powers of the inverse Gaussian distribution. For some particular cases of Stieltjes classes we have found the value of the index of dissimilarity. (C) 2004 Elsevier B.V. All rights reserved.Article Citation Count: 2Uncorrelatedness and correlatedness of powers of random variables(Birkhauser verlag Ag, 2002) Ostrovska, Sofiya; MathematicsLet xi(1),...,xi(n) be random variables and U be a subset of the Cartesian prodnet Z(+)(n), Z(+) being the set of all non-negative integers. The random variables are said to be strictly U-uncorrelated if E(xi(1)(j1) ... xi(n)(jn)) = E(xi(1)(j1)) ... E(xi(n)(jn)) double left right arrow (j(1), ..., j(n)) is an element of U. It is proved that for an arbitrary subset U subset of or equal to Z(+)(n) containing all points with 0 or I non-zero coordinates there exists a collection of n strictly U-uncorrelated random variables.Article Citation Count: 4Uncorrelatedness sets for random variables with given distributions(Amer Mathematical Soc, 2005) Ostrovska, Sofiya; MathematicsLet xi(1) and xi(2) be random variables having finite moments of all orders. The set U(xi(1),xi(2)) := {( j, l) is an element of N-2 : E(xi(1)(j)xi(2)(l)) = E(xi(1)(j)) E(xi(2)(l))} is said to be an uncorrelatedness set of xi(1) and xi(2). It is known that in general, an uncorrelatedness set can be arbitrary. Simple examples show that this is not true for random variables with given distributions. In this paper we present a wide class of probability distributions such that there exist random variables with given distributions from the class having a prescribed uncorrelatedness set. Besides, we discuss the sharpness of the obtained result.Article Citation Count: 1Uncorrelatedness sets of bounded random variables(Academic Press inc Elsevier Science, 2004) Ostrovska, Sofiya; MathematicsAn 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.Article Citation Count: 0Weak uncorrelatedness of random variables(Springer, 2006) Ostrovska, Sofiya; MathematicsNew measures of independence for n random variables, based on their moments, are studied. A scale of degrees of independence for random variables which starts with uncorrelatedness (for n = 2) and finishes at independence is constructed. The scale provides a countable linearly ordered set of measures of independence.