The Euler–Lagrange theory for Schur's Algorithm: Algebraic exposed points

Khrushchev, Sergei

doi:10.1016/j.jat.2005.10.002

Cited by 6 publications

(7 citation statements)

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“…Since r/ √ D is positive on j and j +1 , Lemma 5.5 of [5] (see also [7,Lemma 3.3]) shows that the number of zeros of r in j is odd. The total number of closed arcs j is d + 1 whereas deg(r) d + 1.…”

Section: Theorem 85mentioning

confidence: 99%

“…Another important continuum, the unit ball B of the Hardy algebra H ∞ , can be parameterized by Schur's parameters {a n } n 0 via Wall continued fractions [12] f (z) = a 0 + (1 − |a 0 | 2 )z a 0 z Comparing these two parameterizations, one can transfer theories related with regular continued fractions to those related to Wall continued fractions. In [5] using this approach, we established some properties of algebraic, and in particular quadratic, exposed irrationalities. Continuum B consists of all holomorphic mappings of the unit disc D into itself.…”

Section: Regular Continued Fractionsmentioning

confidence: 99%

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The Euler–Lagrange Theory for Schur's algorithm: Wall pairs

Khrushchev

2006

Journal of Approximation Theory

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Section: Theorem 85mentioning

confidence: 99%

Section: Regular Continued Fractionsmentioning

confidence: 99%

The Euler–Lagrange Theory for Schur's algorithm: Wall pairs

Khrushchev

2006

Journal of Approximation Theory

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“…In [6,7] we used an analogy between regular and Wall continued fractions to extend the Euler-Lagrange theory of real quadratic irrationalities to Schur's algorithm. The central fact of the Euler-Lagrange theory is Lagrange's theorem saying that the regular continued fraction of every real quadratic irrationality is periodic.…”

Section: Motivationmentioning

confidence: 99%

“…Following [6] we call σ ∈ P(T) (equivalently f = f σ ) a quadratic irrationality if f σ is a quadratic irrationality over the commutative ring C[z] of polynomials in z, i.e. if X = f σ is a solution to an irreducible quadratic equation…”

Section: Motivationmentioning

confidence: 99%

Rational compacts and exposed quadratic irrationalities

Khrushchev

2009

Journal of Approximation Theory

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are non-intersecting closed arcs on the unit circle T then their union E is called rational if all harmonic measures ν E (e j ) at ∞ are rational. It is known that the essential support supp ess (σ ) of a periodic measure σ (i.e. the Verblunsky parameters of σ are periodic) is rational and any rational E is a rotation of supp ess (σ ) for a periodic σ . Elementary proofs of these facts are given. The Schur function f of a periodic σ satisfies z A * f 2where the pair (A, B) of polynomials in z is called a Wall pair for σ . Then supp ess (σ ) = {t ∈ T : |b + (t)| 2 4ω}, b + = B + z B * , ω = C(E) 2 deg(b + ) , C(E) being the logarithmic capacity of E. For any monic b with roots on T, b * = b, and ω satisfying 0 < 4ω m 2 b , where m b is the smallest local maximum of |b| on T, there is a Wall pair (A, B) such that b = B + z B * and supp ess (σ ) = {t ∈ T : |b(t)| 2 4ω} for any periodic σ corresponding to (A, B). The solutions to the equation b = B + z B * in B related to Wall pairs are described. As a consequence we obtain the inverse Bernstein inequality for a separable polynomial b with roots on T: inf T |b | 0.5 · m b · deg(b). The inequality is precise. A complete description of essential supports of periodic measures is also given in terms of the phases of Akhiezer's multi-valued analytic function as well as separable monic polynomials related to it with roots on T.

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“…Even in the scalar case p = q = 1, this theme is still far from being exhausted (cf. Boyd [63], Golinskiȋ [107,108], Golinskiȋ/Khrushchev [109], Khrushchev [117][118][119][120][121], Katsnelson [115] and Simon's recent monograph [139] on orthogonal polynomials on the unit circle). Special highlights in this development were produced very recently by D. Alpay and I. Gohberg [27,28] who obtained deep insights into the structure of the Schur parameter sequences of rational functions strictly contractive in the closed unit disk and by V. K. Dubovoj [82] who could characterize the pseudocontinuability of a (scalar) Schur function in terms of its Schur parameters.…”

Section: Introductionmentioning

confidence: 99%

The Schur–Potapov Algorithm for Sequences of Complex p × q Matrices. II

Bogner

Fritzsche

Kirstein

2006

Complex anal.oper.theory

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This is the second and final part of a paper which appeared in a preceding issue of this journal. Herein the methods developed in the earlier sections of this paper are used first to develop a number of applications. A central theme of this paper is to study the interplay between functions from Sp×q,∞(D) and their sequence of SP-parameters. In particular, we describe how certain summability properties of the SP-parameters are expressed in terms of the associated functions from Sp×q,∞(D). As a byproduct of our investigations on the interplay between the SP-algorithm for p × q Schur functions and the SP-algorithm for sequences of complex p×q matrices we present a new approach to the nondegenerate matricial Schur problem. Our method complies with the basic strategy in Schur's classical paper [138] because it does not make use of any tools outside of the theory of Schur functions and Schur sequences. A closer look at the behaviour of distinguished subclasses of Sp×q(D) with respect to the SP-algorithm enables us to handle the corresponding versions of the matricial Schur problem restricted to these subclasses. Mathematics Subject Classification (2000). Primary 30E05, 47A57.Keywords. Schur function, matricial Schur problem, Schur-Potapov matrix polynomials, Arov-Kreȋn matrix polynomials. IntroductionIn the first part [57] of this paper we have established an appropriate adaptation of the Schur-Potapov algorithm for functions belonging to the matricial Schur class S p×q (D) to infinite sequences of complex p × q matrices. Using the fact that the Taylor coefficients sequences of functions from S p×q (D) are infinite p × q Schur sequences we mainly focused on the subclass S p×q,∞ (D) of S p×q (D) which consists of all p × q Schur functions for which the corresponding Taylor coefficient sequences are nondegenerate p × q Schur sequences. Thus, an one-to-one correspondence between infinite nondegenerate p × q Schur sequences and the set of all

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The Euler–Lagrange theory for Schur's Algorithm: Algebraic exposed points

Cited by 6 publications

References 15 publications

The Euler–Lagrange Theory for Schur's algorithm: Wall pairs

The Euler–Lagrange Theory for Schur's algorithm: Wall pairs

Rational compacts and exposed quadratic irrationalities

The Schur–Potapov Algorithm for Sequences of Complex p × q Matrices. II

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