Discrete Sturm-Liouville operators and some problems of function theory

Никишин, Е. М.

doi:10.1007/bf01119188

Cited by 44 publications

(33 citation statements)

References 8 publications

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“…This connection between the spectral theory and analysis is fruitful for various points of view. For example, the scattering problem for a Jacobi matrix can be treated on the basis of strong (or Szegö type) asymptotic results for orthogonal polynomials ( [4], [6], [10]). On the other hand, the perturbation theory gives new results for orthogonal polynomials ( [12]).…”

Section: Introductionmentioning

confidence: 99%

Criterion for the Resolvent Set of Nonsymmetric Tridiagonal Operators

Aptekarev¹,

Kaliaguine²,

Assche³

1995

Proceedings of the American Mathematical Society

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Section: Introductionmentioning

confidence: 99%

Criterion for the Resolvent Set of Nonsymmetric Tridiagonal Operators

Aptekarev¹,

Kaliaguine²,

Assche³

1995

Proceedings of the American Mathematical Society

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“…While it appears that this argument is well known to some experts, I am aware of only two places that it appears explicitly in print: in Nikishin's lovely paper [2] and in my review article on Sturm oscillation theory for the Sturm 200th Birthday Conference [5].…”

Section: Introductionmentioning

confidence: 99%

“…(Indeed, it was this case treated in [2] and [5]; Nikishin doesn't use the terminology "oscillation theorem" but he counts sign changes of certain determinants.) Theorem 1.2.…”

Section: Introductionmentioning

confidence: 99%

On the Removal of Finite Discrete Spectrum by Coefficient Stripping

Simon¹

2011

J. Spectr. Theory

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“…Rakhmanov [8] considered (also implicitly) the case of a measure supported on a finite number of disjoint real intervals; he used the quasiorthogonality property for the associated orthogonal polynomials. Later, Nikishin studied this problem explicitly for the unit circle; this case is closely related to the scattering problem for the second order Sturm-Liouville difference operator [6]. Kaliaguine and Benzine [12] and Kaliaguine [11] obtained asymptotic formulas for the case of one complex curve and one complex arc.…”

Section: §1 Introductionmentioning

confidence: 99%

On the asymptotics of polynomials orthogonal on a system of curves with respect to a measure with discrete part

Kalyagin

Kononova

2010

St. Petersburg Math. J.

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Abstract. Consider an absolutely continuous measure on a system of Jordan arcs and (closed) curves in the complex plane, assuming that this measure satisfies the Szegő condition on its support and that the support of the measure is the boundary of some (multiply connected) domain Ω containing infinity. Adding to the measure a finite number of discrete masses lying in Ω (off the support of the measure), we study the strong asymptotics of the polynomials orthogonal with respect to the perturbed measure. For this, we solve an extremal problem in a certain class of multivalued functions. Our goal is to give an explicit expression for the strong asymptotics on the support of the perturbed measure, as well as on the domain Ω. §1. IntroductionLet σ be a positive measure with compact support in the complex plane. We introduce the sequence of polynomials Q n (z) = z n + · · · orthogonal with respect to the measure σ:A fundamental problem of the theory of orthogonal polynomials is the study of their asymptotic behavior as n → ∞.As usual, by the weak asymptotics of orthogonal polynomials we mean the asymptotics of |Q n (z; σ)| 1/n . The weak asymptotics is closely related to the distribution of zeros of Q n (z) and is determined by the support of the measure σ and by the regularity properties of σ on its support. The main tool of the investigation in this case is the logarithmic potential theory [20].The ratio asymptotics is that of the ratio Q n+1 (z; σ)/Q n (z; σ). This aspect has attracted a lot of attention recently. The main progress was due to the varying weight approach, developed be G. Lopez Lagomasino and his collaborators [9]. On an interval of the real line, the ratio asymptotics of the orthogonal polynomials can be exhibited under various conditions, of which the best by now is due to Rakhmanov : σ > 0 almost everywhere on the interval.The strong asymptotics problem is the problem of uniform asymptotic behavior of the polynomials Q n (z) outside of the support of the measure and on this support. In the classical case where supp(σ) = [−1, 1], the strong asymptotics of orthogonal polynomials was investigated by S. N. Bernstein and G. Szegő [19]. The Bernstein-Szegő theory was extended to the case of the measure supported on a finite system of complex arcs 2000 Mathematics Subject Classification. Primary 42C05; Secondary 30D55, 30E15.

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Discrete Sturm-Liouville operators and some problems of function theory

Cited by 44 publications

References 8 publications

Criterion for the Resolvent Set of Nonsymmetric Tridiagonal Operators

Criterion for the Resolvent Set of Nonsymmetric Tridiagonal Operators

On the Removal of Finite Discrete Spectrum by Coefficient Stripping

On the asymptotics of polynomials orthogonal on a system of curves with respect to a measure with discrete part

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