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 Mat. Sb., 2002, Volume 193, Number 12, Pages 105–133 (Mi msb701)

Approximation properties of the poles of diagonal Padé approximants for certain generalizations of Markov functions

S. P. Suetin

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: A non-linear system of differential equations (‘generalized Dubrovin system") is obtained to describe the behaviour of the zeros of polynomials orthogonal on several intervals that lie in lacunae between the intervals. The same system is shown to describe the dynamical behaviour of zeros of this kind for more general orthogonal polynomials: the denominators of the diagonal Padé approximants of meromorphic functions on a real hyperelliptic Riemann surface.
On the basis of this approach several refinements of Rakhmanov’s results on the convergence of diagonal Padé approximants for rational perturbations of Markov functions are obtained.

DOI: https://doi.org/10.4213/sm701

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English version:
Sbornik: Mathematics, 2002, 193:12, 1837–1866

Bibliographic databases:

UDC: 517.538+517.587
MSC: Primary 41A21, 42C05; Secondary 30F30, 14H40

Citation: S. P. Suetin, “Approximation properties of the poles of diagonal Padé approximants for certain generalizations of Markov functions”, Mat. Sb., 193:12 (2002), 105–133; Sb. Math., 193:12 (2002), 1837–1866

Citation in format AMSBIB
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This publication is cited in the following articles:
1. S. P. Suetin, “The asymptotic behaviour of diagonal Padé approximants for hyperelliptic functions of genus $g=2$”, Russian Math. Surveys, 58:4 (2003), 802–804
2. S. P. Suetin, “Convergence of Chebyshëv continued fractions for elliptic functions”, Sb. Math., 194:12 (2003), 1807–1835
3. S. L. Skorokhodov, “Padé approximation and numerical analysis for the Riemann $\zeta$-function”, Comput. Math. Math. Phys., 43:9 (2003), 1277–1298
4. A. A. Gonchar, S. P. Suetin, “On Padé Approximants of Meromorphic Functions of Markov Type”, Proc. Steklov Inst. Math., 272, suppl. 2 (2011), S58–S95
5. S. L. Skorokhodov, “Methods of analytical continuation of the generalized hypergeometric functions $_pF_{p-1}(a_1,…,a_p;b_1,…,b_{p-1};z)$”, Comput. Math. Math. Phys., 44:7 (2004), 1102–1123
6. S. P. Suetin, “On polynomials orthogonal on several segments with indefinite weight”, Russian Math. Surveys, 60:5 (2005), 991–993
7. S. P. Suetin, “Comparative Asymptotic Behavior of Solutions and Trace Formulas for a Class of Difference Equations”, Proc. Steklov Inst. Math., 272, suppl. 2 (2011), S96–S137
8. S. P. Suetin, “Spectral properties of a class of discrete Sturm–Liouville operators”, Russian Math. Surveys, 61:2 (2006), 365–367
9. S. P. Suetin, “Trace formulae for a class of Jacobi operators”, Sb. Math., 198:6 (2007), 857–885
10. Baratchart, L, “Multipoint Pade approximants to complex Cauchy transforms with polar singularities”, Journal of Approximation Theory, 156:2 (2009), 187
11. S. P. Suetin, “Numerical Analysis of Some Characteristics of the Limit Cycle of the Free van der Pol Equation”, Proc. Steklov Inst. Math., 278, suppl. 1 (2012), S1–S54
12. Derevyagin M., Derkach V., “Convergence of Diagonal Pade Approximants for a Class of Definitizable Functions”, Recent Advances in Operator Theory in Hilbert and Krein Spaces, Operator Theory Advances and Applications, 198, 2010, 97–124
13. 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
14. 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
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