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Mat. Sb., 2016, Volume 207, Number 9, Pages 57–90 (Mi msb8448)  

This article is cited in 6 scientific papers (total in 6 papers)

The Gonchar-Stahl $\rho^2$-theorem and associated directions in the theory of rational approximations of analytic functions

E. A. Rakhmanovab

a University of South Florida, Tampa, FL, USA
b Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

Abstract: The Gonchar-Stahl $\rho^2$-theorem characterizes the rate of convergence of best uniform (Chebyshev) rational approximations (with free poles) for one basic class of analytic functions. The theorem itself, modifications and generalizations of it, methods involved in its proof and other related details constitute an important subfield in the theory of rational approximations of analytic functions and complex analysis.
This paper briefly outlines the essentials of the subfield. The fundamental contributions of A. A. Gonchar and H. Stahl are at the heart of the exposition.
Bibliography: 70 titles.

Keywords: rational approximations, Padé approximants, orthogonal polynomials, equilibrium distributions, stationary compact set, $S$-property.

DOI: https://doi.org/10.4213/sm8448

Full text: PDF file (750 kB)
References: PDF file   HTML file

English version:
Sbornik: Mathematics, 2016, 207:9, 1236–1266

Bibliographic databases:

ArXiv: 1503.06620
UDC: 517.53
MSC: Primary 30E10, 41A20, 41A25; Secondary 41A21
Received: 26.10.2014 and 10.04.2016

Citation: E. A. Rakhmanov, “The Gonchar-Stahl $\rho^2$-theorem and associated directions in the theory of rational approximations of analytic functions”, Mat. Sb., 207:9 (2016), 57–90; Sb. Math., 207:9 (2016), 1236–1266

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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. V. G. Lysov, “Silnaya asimptotika approksimatsii Ermita–Pade dlya sistemy Nikishina s vesami Yakobi”, Preprinty IPM im. M. V. Keldysha, 2017, 085, 35 pp.  mathnet  crossref
    2. V. G. Lysov, D. N. Tulyakov, “On a Vector Potential-Theory Equilibrium Problem with the Angelesco Matrix”, Proc. Steklov Inst. Math., 298 (2017), 170–200  mathnet  crossref  crossref  isi  elib
    3. S. P. Suetin, “Distribution of the zeros of Hermite–Padé polynomials for a complex Nikishin system”, Russian Math. Surveys, 73:2 (2018), 363–365  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    4. E. A. Rakhmanov, “Zero distribution for Angelesco Hermite–Padé polynomials”, Russian Math. Surveys, 73:3 (2018), 457–518  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    5. S. P. Suetin, “On a new approach to the problem of distribution of zeros of Hermite–Padé polynomials for a Nikishin system”, Proc. Steklov Inst. Math., 301 (2018), 245–261  mathnet  crossref  crossref  isi  elib  elib
    6. E. M. Chirka, “Potentials on a compact Riemann surface”, Proc. Steklov Inst. Math., 301 (2018), 272–303  mathnet  crossref  crossref  isi  elib  elib
  • Математический сборник Sbornik: Mathematics (from 1967)
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