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 Mat. Zametki, 2009, Volume 86, Issue 2, Pages 290–303 (Mi mz5262)

On the Existence of Nonlinear Padé–Chebyshev Approximations for Analytic Functions

S. P. Suetin

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: We present examples of two functions that are analytic on the interval $[-1,1]$ and satisfy the condition that, for any $n=2,3,…$, the first of them does not have nonlinear Padé–Chebyshev approximations of type $(n,2)$ and the second function does not have nonlinear Padé–Chebyshev approximations of type $(n,n)$ (i.e., does not have diagonal approximations). Because of the existence criterion for nonlinear Padé–Faber approximations, which is obtained in the present paper, both of these examples follow from the respective well-known V. I. Buslaev counterexamples to the Baker–Graves-Morris conjecture and to the Baker–Gammel–Wills conjecture about the Padé approximations of a power series. In particular, the first of these functions is a rational function of type $(2,3)$, and the second function is also defined by an explicit analytic expression.

Keywords: analytic function, rational function, algebraic function, Padé–Chebyshev approximation, Padé–Faber approximation, Laurent series, Faber series

DOI: https://doi.org/10.4213/mzm5262

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English version:
Mathematical Notes, 2009, 86:2, 264–275

Bibliographic databases:

UDC: 517.538
Revised: 31.10.2008

Citation: S. P. Suetin, “On the Existence of Nonlinear Padé–Chebyshev Approximations for Analytic Functions”, Mat. Zametki, 86:2 (2009), 290–303; Math. Notes, 86:2 (2009), 264–275

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/mz5262
• https://doi.org/10.4213/mzm5262
• http://mi.mathnet.ru/eng/mz/v86/i2/p290

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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. L. A. Knizhnerman, “Padé–Faber Approximation of Markov Functions on Real-Symmetric Compact Sets”, Math. Notes, 86:1 (2009), 81–92
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. A. Gonchar, E. A. Rakhmanov, S. P. Suetin, “Padé–Chebyshev approximants of multivalued analytic functions, variation of equilibrium energy, and the $S$-property of stationary compact sets”, Russian Math. Surveys, 66:6 (2011), 1015–1048
4. Bosuwan N., “On Montessus de Ballore'S Theorem For Nonlinear Pade-Orthogonal Approximants”, Jaen J. Approx., 8:2 (2016), 151–173
5. A. P. Starovoitov, E. P. Kechko, “On Some Properties of Hermite–Padé Approximants to an Exponential System”, Proc. Steklov Inst. Math., 298 (2017), 317–333
6. Bosuwan N., “On Montessus de Ballore'S Theorem For Simultaneous Pade-Faber Approximants”, Demonstr. Math., 51:1 (2018), 45–61
7. N. Bosuwan, “Convergence of Row Sequences of Simultaneous Padé–Faber Approximants”, Math. Notes, 103:5 (2018), 683–693
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