Khrushchev, Sergey
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Khrushchev, S
Khrushchev, S.
Khrushchev, SV
S.,Khrushchev
K.,Sergey
K., Sergey
Sergey, Khrushchev
S., Khrushchev
Khrushchev,S.
Khrushchev, Sergey
Khrushchev, S.
Khrushchev, SV
S.,Khrushchev
K.,Sergey
K., Sergey
Sergey, Khrushchev
S., Khrushchev
Khrushchev,S.
Khrushchev, Sergey
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Scholarly Output
16
Articles
5
Citation Count
52
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0
16 results
Scholarly Output Search Results
Now showing 1 - 10 of 16
Article Citation Count: 25Classification theorems for general orthogonal polynomials on the unit circle(Academic Press inc Elsevier Science, 2002) Khrushchev, Sergey; MathematicsThe 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).Conference Object Citation Count: 5Continued fractions and orthogonal polynomials on the unit circle(Elsevier Science Bv, 2005) Khrushchev, Sergey; MathematicsThis 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.Book Part Citation Count: 0Orthogonal polynomials on the unit circle(Cambridge Univ Press, 2008) Khrushchev, Sergey; Mathematics[No Abstract Available]Book Part Citation Count: 0Orthogonal polynomials(Cambridge Univ Press, 2008) Khrushchev, Sergey; Mathematics[No Abstract Available]Book Part Citation Count: 0Continued fractions: Euler(Cambridge Univ Press, 2008) Khrushchev, Sergey; Mathematics[No Abstract Available]Review Citation Count: 5A recovery of Brouncker's proof for the quadrature continued fraction(Univ Autonoma Barcelona, 2006) Khrushchev, Sergey; Mathematics350 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.Article Citation Count: 5The Euler-Lagrange theory for Schur's Algorithm: Algebraic exposed points(Academic Press inc Elsevier Science, 2006) Khrushchev, Sergey; MathematicsIn 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.Conference Object Citation Count: 0The great theorem of AA Markoff and Jean Bernoulli sequences(Elsevier Science Bv, 2010) Khrushchev, Sergey; MathematicsA proof of Markoff's Great Theorem on the Lagrange spectrum using continued fractions is sketched. Markoff's periods and jean Bernoulli sequence(1) are used to obtain a simple algorithm for the computation of the Lagrange spectrum below 3. (C) 2009 Elsevier B.V. All rights reserved.Article Citation Count: 1Turan measures(Academic Press inc Elsevier Science, 2003) Khrushchev, Sergey; MathematicsA 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 Count: 7Cesaro asymptotics for orthogonal polynomials on the unit circle and classes of measures(Academic Press inc Elsevier Science, 2002) Khrushchev, Sergey; Khrushchev, S; MathematicsThe 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).
