Browsing by Author "Khrushchev, S"

Now showing 1 - 6 of 6

Citation - WoS: 7
Citation - Scopus: 9
Cesaro Asymptotics for Orthogonal Polynomials on the Unit Circle and Classes of Measures
(Academic Press inc Elsevier Science, 2002) Golinskii, L; Khrushchev, S
The 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).
Citation - WoS: 4
Citation - Scopus: 2
Continued Fractions and Orthogonal Polynomials on the Unit Circle
(Elsevier Science Bv, 2005) Khrushchev, S
This survey is written to stress the role of continued fractions in the theory of orthogonal polynomials on the line and on the circle. We follow the historical development of the subject, which opens many interesting relationships of orthogonal polynomials to other important branches of mathematics. At the end we present a new formula for orthogonal polynomials on the real line, the Leganes formula, [GRAPHICS] which is a correct analogue of the corresponding formula on the unit circle. This formula is applied to obtain a recent result by Simon. (c) 2004 Elsevier B.V. All rights reserved.
Citation - WoS: 5
Citation - Scopus: 5
The Euler-Lagrange Theory for Schur's Algorithm: Algebraic Exposed Points
(Academic Press inc Elsevier Science, 2006) Khrushchev, S
In 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.
Citation - WoS: 4
Citation - Scopus: 3
The Euler-Lagrange Theory for Schur's Algorithm: Wall Pairs
(Academic Press inc Elsevier Science, 2006) Khrushchev, S
This 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.
Citation - WoS: 5
Citation - Scopus: 4
A Recovery of Brouncker's Proof for the Quadrature Continued Fraction
(Univ Autonoma Barcelona, 2006) Khrushchev, S
350 years ago in Spring of 1655 Sir William Brouncker on a request by John Wallis obtained a beautiful continued fraction for 4/pi. Brouncker never published his proof. Many sources on the history of Mathematics claim that this proof was lost forever. In this paper we recover the original proof from Wallis' remarks presented in his "Arithmetica Infinitorum". We show that Brouncker's and Wallis' formulas can be extended to MacLaurin's sinusoidal spirals via related Euler's products. We derive Ramanujan's formula from Euler's formula and, by using it, then show that numerators of convergents of Brouncker's continued fractions coincide tip to a rotation with Wilson's orthogonal polynomials corresponding to the parameters a = 0, b = 1/2, c = d = 1/4.
Citation - WoS: 1
Citation - Scopus: 2
Turan measures
(Academic Press inc Elsevier Science, 2003) Khrushchev, S
A 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.