##
**The Gonchar-Stahl \(\rho^2\)-theorem and associated directions in the theory of rational approximations of analytic functions.**
*(English.
Russian original)*
Zbl 1361.30061

Sb. Math. 207, No. 9, 1236-1266 (2016); translation from Mat. Sb. 207, No. 9, 57-90 (2016).

Let \(f\) be an analytic function on a continuum \(E\) (with \(\hat{\mathbb C}\setminus{E}\) connected, \(\hat{\mathbb C}:=\mathbb C\cup\{\infty\}\)) which admits analytic continuation along any path in a domain \(\mathcal D\) with \(E\subset\mathcal D\) and \(\mathrm{cap}(\hat{\mathbb C}\setminus\mathcal D)=0\). Further, let \(\rho_n(f,E)\) be the supremum norm of the best rational approximation of \(f\), that is,
\[
\rho_n(f,E):=\min\sup_{z\in{E}}|f(z)-r(z)|,
\]
where the minimum is taken over all rational functions \(r=P_n/Q_n\) and \(P_n,Q_n\) polynomials of degree at most \(n\). Finally, define
\[
\rho(f,E):=\inf\mathrm e^{-1/\mathrm{cap}(E,F)},
\]
where \(\mathrm{cap}(E,F)\) is the capacity of the condenser \((E,F)\) and the infimum is taken over all sets \(F=\partial\Omega\) with \(E\subset\Omega\) and \(f\) holomorphic on \(\Omega\). One of the central results in the theory of rational approximation of analytic functions is the so-called Gonchar-Stahl \(\rho^2\)-theorem, which says that
\[
\lim_{n\to\infty}\rho_n(f,E)^{1/n}=\rho(f,E)^2.
\]
In this survey article, the author gives a thorough insight into the theory of rational approximation of analytic functions, both from a historical point of view starting from a well-known theorem of Walsh in 1930’s and also by describing the various methods (which are interesting for themselves) used in the proof. Furthermore, the connection to Padé approximation of functions with branch points and some generalizations and conjectures are discussed.

Reviewer: Klaus Schiefermayr (Wels)

### MSC:

30E10 | Approximation in the complex plane |

30C85 | Capacity and harmonic measure in the complex plane |

31A15 | Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions |

41A20 | Approximation by rational functions |

41A25 | Rate of convergence, degree of approximation |

41A21 | Padé approximation |

PDFBibTeX
XMLCite

\textit{E. A. Rakhmanov}, Sb. Math. 207, No. 9, 1236--1266 (2016; Zbl 1361.30061); translation from Mat. Sb. 207, No. 9, 57--90 (2016)

### References:

[1] | Aptekarev A. I. 2002 Sharp constants for rational approximations of analytic functions Mat. Sb.193 3-72 |

[2] | Aptekarev A. I. 2008 Asymptotics of Hermite-Padé approximants for two functions with branch points Dokl. Ross. Akad. Nauk422 443-445 · Zbl 1181.30022 |

[3] | Aptekarev A. I., Kuijlaars A. B. J. and Assche W. Van 2008 Asymptotics of Hermite-Padé rational approximants for two analytic functions with separated pairs of branch points (case of genus Int. Math. Res. Pap. IMRP2008 · Zbl 1156.41004 |

[4] | Aptekarev A. I., Buslaev V. I., Martínez-Finkelshtein A. and Suetin S. P. 2011 Padé approximants, continued fractions, and orthogonal polynomials Uspekhi Mat. Nauk66 37-122 · Zbl 1242.41014 |

[5] | Aptekarev A., Nevai P. and Totik V. 2014 In memoriam: Herbert Stahl August 3, 1942–April 22, 2013 J. Approx. Theory183 A1-A26 · Zbl 1288.01029 |

[6] | Aptekarev A. I. and Yattselev M. L. 2015 Padé approximants for functions with branch points – strong asymptotics of Nuttall-Stahl polynomials Acta Math.215 217-280 · Zbl 1339.41020 |

[7] | Baker G. A., Jr. Jr. and Graves-Morris P. 1981 Encyclopedia Math. Appl.13 (Reading, Mass.: Addison-Wesley Publishing Co.) |

[8] | Baratchart L. and Yattselev M. 2010 Convergent interpolation to Cauchy integrals over analytic arcs with Jacobi-type weights Int. Math. Res. Not. IMRN2010 4211-4275 · Zbl 1215.41005 |

[9] | Baratchart L., Stahl H. and Yattselev M. 2012 Weighted extremal domains and best rational approximation Adv. Math.229 357-407 · Zbl 1232.41014 |

[10] | Beckermann B., Kalyagin V., Matos A. C. and Wielonsky F. 2013 Equilibrium problems for vector potentials with semidefinite interaction matrices and constrained masses Constr. Approx.37 101-134 · Zbl 1261.31001 |

[11] | Buslaev V. I. and Suetin S. P. 2014 Existence of compact sets with minimum capacity in problems of rational approximation of multivalued analytic functions Uspekhi Mat. Nauk69 169-170 · Zbl 1290.31002 |

[12] | Buslaev V. I. and Suetin S. P. 2014 An extremal problem in potential theory Uspekhi Mat. Nauk69 157-158 |

[13] | Buslaev V. I. 2015 Convergence of Mat. Sb.206 5-30 |

[14] | Buslaev V. I. 2015 Capacity of a compact set in a logarithmic potential field Proc. Steklov Inst. Math.290 254-271 |

[15] | Buslaev V. I. and Suetin S. P. 2015 On equilibrium problems related to the distribution of zeros of the Hermite-Padé polynomials Proc. Steklov Inst. Math.290 272-279 |

[16] | Buslaev V. I. and Suetin S. P. 2016 On the existence of compacta of minimal capacity in the theory of rational approximation of multi-valued analytic functions J. Approx. Theory206 48-67 · Zbl 1347.41014 |

[17] | (Chebyshev) P. Tchébycheff 1858 Sur les fractions continues J. Math. Pures Appl. Sér. 23 289-323 |

[18] | no A. Dea, Huybrechs D. and Kuijlaars A. B. J. 2010 Asymptotic zero distribution of complex orthogonal polynomials associated with Gaussian quadrature J. Approx. Theory162 2202-2224 · Zbl 1223.41017 |

[19] | Erokhin V. D. 1959 On the best approximation of analytic functions by rational fractions with free poles Dokl. Akad. Nauk SSSR128 29-32 |

[20] | Hardy A. and Kuijlaars A. B. J. 2013 Weakly admissible vector equilibrium problems J. Approx. Theory170 44-58 · Zbl 1281.31002 |

[21] | (Gonchar) A. A. Gončar 1967 On the rapidity of rational approximation of continuous functions with characteristic singularities Mat. Sb.73(115) 630-638 |

[22] | Gonchar A. A. 1972 A local condition of single-valuedness of analytic functions Mat. Sb.89(131) 148-164 |

[23] | (Gonchar) A. A. Gončar 1974 The rate of rational approximation and the property of single-valuedness of an analytic function in the neighborhood of an isolated singular point Mat. Sb.94(136) 265-282 |

[24] | (Gonchar) A. A. Gončar 1974 A local condition for the single-valuedness of analytic functions of several variables Mat. Sb.93(135) 296-313 |

[25] | (Gonchar) A. A. Gončar 1978 On the speed of rational approximation of some analytic functions Mat. Sb.105(147) 147-163 |

[26] | Gonchar A. A. 1978 5.6. Rational approximation of analytic functions J. Soviet Math.81 182-185 |

[27] | (Gonchar) A. A. Gončar and Lagomasino G. López 1978 On Markov’s theorem for multipoint Padé approximants Mat. Sb.105(147) 512-524 |

[28] | Gonchar A. A. and Rakhmanov E. A. 1981 On convergence of simultaneous Padé approximants for systems of functions of Markov type Proc. Steklov Inst. Math.157 31-48 · Zbl 0492.41027 |

[29] | Gonchar A. A. 1984 On the degree of rational approximation of analytic functions Proc. Steklov Inst. Math.166 52-60 · Zbl 0575.30035 |

[30] | Gonchar A. A. 1984 Rational approximation of analytic functions Linear and complex analysis problem book1043 471-474 |

[31] | Gonchar A. A. and Rakhmanov E. A. 1984 Equilibrium measure and the distribution of zeros of extremal polynomials Mat. Sb.125(167) 117-127 |

[32] | Gonchar A. A. and Rakhmanov E. A. 1985 On the equilibrium problem for vector potentials Uspekhi Mat. Nauk40 155-156 |

[33] | Gonchar A. A. 1987 Rational approximations of analytic functions Proceedings of the International Congress of Mathematicians147 739-748 |

[34] | Gonchar A. A. and Rakhmanov E. A. 1987 Equilibrium distributions and degree of rational approximation of analytic functions Mat. Sb.134(176) 306-352 · Zbl 0645.30026 |

[35] | Gonchar A. A., Rakhmanov E. A. and Sorokin V. N. 1997 Hermite-Padé approximants for systems of Markov-type functions Mat. Sb.188 33-58 |

[36] | Gonchar A. A. 2003 Rational approximation of analytic functions Proc. Steklov Inst. Math.1 83-106 |

[37] | Lagomasino G. López, Martínez-Finkelshtein A., Nevai P. and Saff E. B. 2013 Andrei Aleksandrovich Gonchar, November 21, 1931–October 10, 2012 J. Approx. Theory172 A1-A13 · Zbl 1279.01042 |

[38] | Komlov A. V., Kruzhilin N. G., Palvelev R. V. and Suetin S. P. 2016 Convergence of Shafer quadratic approximants Uspekhi Mat. Nauk71 205-206 · Zbl 1350.41003 |

[39] | Kuijlaars A. B. J. and Silva G. L. F. 2015 J. Approx. Theory191 1-37 · Zbl 1314.31006 |

[40] | Magnus A. P. 1988 On the use of Carathéodory-Fejér method for investigating Nonlinear numerical methods and rational approximation43 105-132 |

[41] | Markoff A. 1895 Deux démonstrations de la convergence de certaines fractions continues Acta Math.19 93-104 · JFM 26.0234.01 |

[42] | Martínez-Finkelshtein A. and Rakhmanov E. 2011 Critical measures, quadratic differentials, and weak limits of zeros of Stieltjes polynomials Comm. Math. Phys.302 53-111 · Zbl 1226.30005 |

[43] | Martínez-Finkelshtein A., Rakhmanov E. A. and Suetin S. P. 2012 Heine, Hilbert, Padé, Riemann, and Stieltjes: John Nuttall’s work 25 years later Recent advances in orthogonal polynomials, special functions, and their applications578 165-193 · Zbl 1318.42033 |

[44] | Mergelyan S. N. 1961 On some results in the theory of uniform and best approximations by polynomials and rational functions J. L. Walsh, Interpolation and approximation by rational functions in the complex domain 461-499 |

[45] | Newman D. J. 1964 Rational approximation to Michigan Math. J.11 11-14 · Zbl 0138.04402 |

[46] | Nuttall J. and Singh R. S. 1977 Orthogonal polynomials and Padé approximants associated with a system of arcs J. Approximation Theory21 1-42 · Zbl 0355.30004 |

[47] | Nuttall J. 1984 Asymptotics of diagonal Hermite-Padé polynomials J. Approx. Theory42 299-386 · Zbl 0565.41015 |

[48] | Parfenov O. G. 1986 Estimates of the singular numbers of the Carleson imbedding operator Mat. Sb.131(173) 501-518 |

[49] | Perevoznikova E. and Rakhmanov E. 1994 Variations of the equilibrium energy and |

[50] | Pommerenke Ch. 1973 Padé approximants and convergence in capacity J. Math. Anal. Appl.41 775-780 · Zbl 0256.30037 |

[51] | Prokhorov V. A. 1993 On a theorem of Adamyan, Arov, and Kreĭn Mat. Sb.184 89-104 |

[52] | Prokhorov V. A. 1993 Rational approximation of analytic functions Mat. Sb.184 3-32 |

[53] | Prokhorov V. A. 1994 On the degree of rational approximation of meromorphic functions Mat. Sb.185 3-26 |

[54] | Rakhmanov E. A. and Suetin S. P. 2013 The distribution of the zeros of the Hermite-Padé polynomials for a pair of functions forming a Nikishin system Mat. Sb.204 115-160 · Zbl 1288.26010 |

[55] | Rakhmanov E. A. 2012 Orthogonal polynomials and Recent advances in orthogonal polynomials, special functions, and their applications578 195-239 · Zbl 1318.30056 |

[56] | Stahl H. 1985 Extremal domains associated with an analytic function. I Complex Variables Theory Appl.4 311-324 · Zbl 0542.30027 |

[57] | Stahl H. 1985 Extremal domains associated with an analytic function. II Complex Variables Theory Appl.4 325-338 · Zbl 0542.30028 |

[58] | Stahl H. 1985 The structure of extremal domains associated with an analytic function Complex Variables Theory Appl.4 339-354 · Zbl 0542.30029 |

[59] | Stahl H. 1986 Orthogonal polynomials with complex-valued weight function. I Constr. Approx.2 225-240 · Zbl 0592.42016 |

[60] | Stahl H. 1986 Orthogonal polynomials with complex-valued weight function. II Constr. Approx.2 241-251 · Zbl 0606.42021 |

[61] | Stahl H. 1992 Best uniform rational approximation of Mat. Sb.183 85-118 |

[62] | Stahl H. and Totik V. 1992 Encyclopedia Math. Appl.43 (Cambridge: Cambridge Univ. Press) |

[63] | Strebel K. 1984 Ergeb. Math. Grenzgeb. (3)5 (Berlin: Springer-Verlag) |

[64] | Suetin S. P. 2015 Distribution of the zeros of Padé polynomials and analytic continuation Uspekhi Mat. Nauk70 121-174 |

[65] | Poussin Ch.-J. De la Vallee 1911 Sur les polynomes d’approximation à une variable complexe Bull. Acad. Roy. de Belgique Cl. Sci.3 199-211 |

[66] | Varga R. S., Ruttan A. and Carpenter A. J. 1991 Numerical results on best uniform rational approximation of Mat. Sb.182 1523-1541 · Zbl 0739.65010 |

[67] | Vyacheslavov N. S. 1974 Approximation of the function Mat. Zametki16 163-171 · Zbl 0314.41007 |

[68] | Walsh J. L. 1934 On approximation to an analytic function by rational functions of best approximation Math. Z.38 163-176 · Zbl 0008.11203 |

[69] | Walsh J. L. 1960 Amer. Math. Soc. Colloq. Publ.XX (Providence, RI: Amer. Math. Soc.) |

[70] | Walsh J. L. 1974 Padé approximants as limits of rational functions of best approximation, real domain J. Approximation Theory11 225-230 · Zbl 0306.41010 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.