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 Sovrem. Probl. Mat., 2006, Issue 6, Pages 3–74 (Mi spm9)

Comparative Asymptotic Behavior of Solutions and Trace Formulas for a Class of Difference Equations

S. P. Suetin

Abstract: Properties of Jacobi operators generated by Markov functions are studied. The main results refer to the case where the support of the corresponding spectral measure $\mu$ consists of several intervals of the real line. In this class of operators, a comparative asymptotic formula for two solutions of the corresponding difference equation, polynomials orthogonal with respect to the measure $\mu$ and functions of the second kind (Weyl solutions) is found. Asymptotic trace formulas for the coefficients $a_n$ and $b_n$ in this difference equation are obtained. The derivation of the asymptotic formulas is based on standard techniques for studying the asymptotic properties of polynomials orthogonal on several intervals and consists in reducing the asymptotic problem to investigating properties of solutions to the Nuttall singular integral equation.

DOI: https://doi.org/10.4213/spm9

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English version:
Proceedings of the Steklov Institute of Mathematics (Supplementary issues), 2011, 272, suppl. 2, S96–S137

Bibliographic databases:

UDC: 517.53+517.984+517.962

Citation: S. P. Suetin, “Comparative Asymptotic Behavior of Solutions and Trace Formulas for a Class of Difference Equations”, Sovrem. Probl. Mat., 6, Steklov Math. Institute of RAS, Moscow, 2006, 3–74; Proc. Steklov Inst. Math., 272, suppl. 2 (2011), S96–S137

Citation in format AMSBIB
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• https://doi.org/10.4213/spm9
• http://mi.mathnet.ru/eng/spm/v6/p3

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. S. P. Suetin, “Trace formulae for a class of Jacobi operators”, Sb. Math., 198:6 (2007), 857–885
2. A. I. Aptekarev, V. I. Buslaev, A. Martínez-Finkelshtein, S. P. Suetin, “Padé approximants, continued fractions, and orthogonal polynomials”, Russian Math. Surveys, 66:6 (2011), 1049–1131
3. A. V. Komlov, S. P. Suetin, “Widom's formula for the leading coefficient of a polynomial which is orthonormal with respect to a varying weight”, Russian Math. Surveys, 67:1 (2012), 183–185
4. A. V. Komlov, S. P. Suetin, “An asymptotic formula for polynomials orthonormal with respect to a varying weight”, Trans. Moscow Math. Soc., 73 (2012), 139–159
5. E. A. Rakhmanov, S. P. Suetin, “The distribution of the zeros of the Hermite-Padé polynomials for a pair of functions forming a Nikishin system”, Sb. Math., 204:9 (2013), 1347–1390
6. A. V. Komlov, S. P. Suetin, “An asymptotic formula for polynomials orthonormal with respect to a varying weight. II”, Sb. Math., 205:9 (2014), 1334–1356
7. N. R. Ikonomov, R. K. Kovacheva, S. P. Suetin, “Nuttall's integral equation and Bernshtein's asymptotic formula for a complex weight”, Izv. Math., 79:6 (2015), 1215–1234
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