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Article Citation - WoS: 1Citation - Scopus: 2Turan measures(Academic Press inc Elsevier Science, 2003) Khrushchev, SA probability measure a on the unit circle T is called a Turan measure if any point of the open unit disc D is a limit point of zeros of the orthogonal polynomials associated to a. We show that many classes of measures, including Szego measures, measures with absolutely convergent series of their parameters, absolutely continuous measures with smooth densities, contain Turan measures. (C) 2003 Elsevier Science (USA). All rights reserved.Article Citation - WoS: 4Citation - Scopus: 3The Euler-Lagrange Theory for Schur's Algorithm: Wall Pairs(Academic Press inc Elsevier Science, 2006) Khrushchev, SThis paper develops a techniques of Wall pairs for the study of periodic exposed quadratic irrationalities in the unit ball of the Hardy algebra. (C) 2005 Elsevier Inc. All rights reserved.Article Citation - WoS: 7Citation - Scopus: 9Cesaro Asymptotics for Orthogonal Polynomials on the Unit Circle and Classes of Measures(Academic Press inc Elsevier Science, 2002) Golinskii, L; Khrushchev, SThe convergence in L-2(T) of the even approximants of the Wall continued fractions is extended to the Cesaro-Nevai class CN, which is defined as the class of probability measures sigma with lim(n-->infinity) 1/n Sigma(k=0)(n-1) \a(k)\ = 0, (a(n))(ngreater than or equal to0) being the Geronimus parameters of sigma. We show that CN contains universal measures, that is, probability measures for which the sequence (\phi(n)\(2) dsigma)(ngreater than or equal to0) is dense in the set of all probability measures equipped with the weak-* topology. We also consider the "opposite" Szego class which consists of measures with Sigma(n=0)(infinity) (1-\a(n)\(2))(1/2) < infinity and describe it in terms of Hessenberg matrices. (C) 2002 Elsevier Science (USA).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: 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: 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: 5Citation - Scopus: 5The Euler-Lagrange Theory for Schur's Algorithm: Algebraic Exposed Points(Academic Press inc Elsevier Science, 2006) Khrushchev, SIn this paper the ideas of Algebraic Number Theory are applied to the Theory of Orthogonal polynomials for algebraic measures. The transferring tool are Wall continued fractions. It is shown that any set of closed arcs on the circle supports a quadratic measure and that any algebraic measure is either a Szego measure or a measure supported by a proper subset of the unit circle consisting of a finite number of closed arcs. Singular parts of algebraic measures are finite sums of point masses. (C) 2005 Elsevier Inc. All rights reserved.Article Citation - WoS: 26Citation - Scopus: 26Classification Theorems for General Orthogonal Polynomials on the Unit Circle(Academic Press inc Elsevier Science, 2002) Khrushchev, SVThe set P of all probability measures a on the unit circle T splits into three disjoint subsets depending on properties of the derived set of {\phi(n)\(2) dsigma}(ngreater than or equal to0), denoted by Lim(sigma). Here {phi(n)}(ngreater than or equal to0) are orthogonal polynomials in L-2(dsigma). The first subset is the set of Rakhmanov measures, i.e., of sigma is an element of P with {m} = Lim(sigma), m being the normalized (m(T) = 1) Lebesgue measure on T. The second subset Mar(T) consists of Markoff measures, i.e., of sigma is an element of P with m is not an element of Lim(sigma), and is in fact the subject of study for the present paper. A measure sigma, belongs to Mar(T) iff there are epsilon > 0 and l > 0 such that sup{\a(n+j)\: 0 less than or equal to j less than or equal to l) > epsilon, n = 0, 1, 2,..., {a(n)} is the Geronimus parameters (= reflection coefficients) of sigma. We use this equivalence to describe the asymptotic behavior of the zeros of the corresponding orthogonal polynomials (see Theorem G). The third subset consists of sigma is an element of P with {m} not subset of or equal toLim(sigma). We show that sigma is ratio asymptotic iff either sigma is a Rakhmanov measure or sigma satisfies the Lopez condition (which implies sigma is an element of Mar(T)). Measures sigma satisfying Lim(sigma) = {v} (i.e., weakly asymptotic measures) are also classified. Either v is the sum of equal point masses placed at the roots of z(n) = lambda, lambda is an element of T, n = 1, 2,..., or v is the equilibrium measure (with respect to the logarithmic kernel) for the inverse image under an m-preserving endomorphism z -->z(n), = 1, 2,..., of a closed arc J (including J = T) with removed open concentric are J(0) (including J(0) = empty set). Next, weakly asymptotic measures are completely described in terms of their Geronimus parameters. Finally, we obtain explicit formulae for the parameters of the equilibrium measures v and show that these measures satisfy {v} = Lim(v). (C) 2002 Elsevier Science (USA).
