Abstract:
The Gonchar-Stahl ρ2-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.
Citation:
E. A. Rakhmanov, “The Gonchar-Stahl ρ2-theorem and associated directions in the theory of rational approximations of analytic functions”, Sb. Math., 207:9 (2016), 1236–1266
\Bibitem{Rak16}
\by E.~A.~Rakhmanov
\paper The Gonchar-Stahl $\rho^2$-theorem and associated directions in the theory of rational approximations of analytic functions
\jour Sb. Math.
\yr 2016
\vol 207
\issue 9
\pages 1236--1266
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S. P. Suetin, “O skalyarnykh podkhodakh k izucheniyu predelnogo raspredeleniya nulei mnogochlenov Ermita–Pade dlya sistemy Nikishina”, UMN, 80:1(481) (2025), 85–152
Lloyd N. Trefethen, “Polynomial and rational convergence rates for Laplace problems on planar domains”, Proc. R. Soc. A., 480:2295 (2024)
A. P. Magnus, J. Meinguet, “Strong asymptotics of the best rational approximation to the exponential function on a bounded interval”, Sb. Math., 215:12 (2024), 1666–1719
Lloyd N. Trefethen, “Numerical analytic continuation”, Japan J. Indust. Appl. Math., 40:3 (2023), 1587
L. N. Trefethen, Yu. Nakatsukasa, J. A. C. Weideman, “Exponential node clustering at singularities for rational approximation, quadrature, and PDEs”, Numer. Math., 147:1 (2021), 227–254
S. P. Suetin, “Distribution of the zeros of Hermite–Padé polynomials for a complex Nikishin system”, Russian Math. Surveys, 73:2 (2018), 363–365
E. A. Rakhmanov, “Zero distribution for Angelesco Hermite–Padé polynomials”, Russian Math. Surveys, 73:3 (2018), 457–518
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
E. M. Chirka, “Potentials on a compact Riemann surface”, Proc. Steklov Inst. Math., 301 (2018), 272–303
V. G. Lysov, “Silnaya asimptotika approksimatsii Ermita–Pade dlya sistemy Nikishina s vesami Yakobi”, Preprinty IPM im. M. V. Keldysha, 2017, 085, 35 pp.
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
Vladimir Genrikhovich Lysov, “On Hermite-Pade approximants for the product of two logarithms”, KIAM Prepr., 2017, no. 141, 1
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