Cesàro Asymptotics for Orthogonal Polynomials on the Unit Circle and Classes of Measures

Golinskiĭ, Leonid; Khrushchev, Sergei

doi:10.1006/jath.2001.3655

Cited by 30 publications

(14 citation statements)

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“…We refer the interested reader, to [22] for a simple proof as well as for a discussion of further results, such as presented in [7] and [8]. In fact we only use Levinson's theorem for the special case M(y)=1/y, when the proof is easy.…”

Section: Then For Every D > 0 There Is a Positive Constant C(d) Such mentioning

confidence: 98%

“…It is shown in [22] that the orthogonal polynomials of any opposite Szegő measure satisfy asymptotic formulae, which, of course, are not of ratio type. The importance of the class OS is related with the fact that it includes measures related with classical ''discrete'' orthogonal polynomials.…”

mentioning

confidence: 96%

“…Another important subclass of Markoff measures has been introduced in [22]. A measure s is said to belong to OS (=s is an opposite Szegő measure) if the Geronimus parameters {a n } of s satisfy…”

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confidence: 99%

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Classification Theorems for General Orthogonal Polynomials on the Unit Circle

Khrushchev¹

2002

Journal of Approximation Theory

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The set P of all probability measures s on the unit circle T splits into three disjoint subsets depending on properties of the derived set of {|j n | 2 ds} n \ 0 , denoted by Lim(s). Here {j n } n \ 0 are orthogonal polynomials in L 2 (ds). The first subset is the set of Rakhmanov measures, i.e., of s ¥ P with {m}=Lim(s), m being the normalized (m(T)=1) Lebesgue measure on T. The second subset Mar(T) consists of Markoff measures, i.e., of s ¥ P with m¨Lim(s), and is in fact the subject of study for the present paper. A measure s, belongs to Mar(T) iff there are e > 0 and l > 0 such that sup{|a n+j |: 0 [ j [ l} > e, n=0, 1, 2, ..., {a n } is the Geronimus parameters (=reflection coefficients) of s. 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 s ¥ P with {m} v Lim(s). We show that s is ratio asymptotic iff either s is a Rakhmanov measure or s satisfies the Ló pez condition (which implies s ¥ Mar(T)). Measures s satisfying Lim(s)={n} (i.e., weakly asymptotic measures) are also classified. Either n is the sum of equal point masses placed at the roots of z n =l, l ¥ T, n=1, 2, ..., or n is the equilibrium measure (with respect to the logarithmic kernel) for the inverse image under an m-preserving endomorphism z Q z n , n=1, 2, ..., of a closed arc J (including J=T) with removed open concentric arc J 0 (including J 0 ="). 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 n and show that these measures satisfy {n}=Lim(n).

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Section: Then For Every D > 0 There Is a Positive Constant C(d) Such mentioning

confidence: 98%

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confidence: 96%

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Classification Theorems for General Orthogonal Polynomials on the Unit Circle

Khrushchev¹

2002

Journal of Approximation Theory

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“…is the Blaschke product of zeros and η n (θ ) is defined by b n (e iθ ) = e iη n (θ) (2.31) then, as shown in [14],…”

Section: (218)mentioning

confidence: 99%

Asymptotics of the L2 norm of derivatives of OPUC

Martínez–Finkelshtein

Simon

2011

Journal of Approximation Theory

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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: 96%

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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Cesàro Asymptotics for Orthogonal Polynomials on the Unit Circle and Classes of Measures

Cited by 30 publications

References 15 publications

Classification Theorems for General Orthogonal Polynomials on the Unit Circle

Classification Theorems for General Orthogonal Polynomials on the Unit Circle

Asymptotics of the L2 norm of derivatives of OPUC

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

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