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

E. A. Rakhmanov

Sbornik. Mathematics, Tome 207 (2016) no. 9, pp. 1236-1266

Résumé

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.

Détail

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

Bibliographie

[1] A. I. Aptekarev, “Sharp constants for rational approximations of analytic functions”, Sb. Math., 193:1 (2002), 1–72 | DOI | DOI | MR | Zbl | Zbl

[2] A. I. Aptekarev, “Asymptotics of Hermite–Padé approximants for two functions with branch points”, Dokl. Math., 78:2 (2008), 717–719 | DOI | MR | Zbl | Zbl

[3] A. I. Aptekarev, A. B. J. Kuijlaars, W. Van Assche, “Asymptotics of Hermite–Padé rational approximants for two analytic functions with separated pairs of branch points (case of genus $0$)”, Int. Math. Res. Pap. IMRP, 2008:4 (2008), rpm007, 128 pp. | DOI | MR | Zbl

[4] 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 | DOI | DOI | MR | Zbl | Zbl

[5] A. Aptekarev, P. Nevai, V. Totik, “In memoriam: Herbert Stahl August 3, 1942–April 22, 2013”, J. Approx. Theory, 183 (2014), A1–A26 | DOI | MR | Zbl

[6] A. I. Aptekarev, M. L. Yattselev, “Padé approximants for functions with branch points – strong asymptotics of Nuttall–Stahl polynomials”, Acta Math., 215:2 (2015), 217–280 | DOI | MR | Zbl

[7] G. A. Baker, Jr., P. Graves-Morris, Padé approximants, Part I. Basic theory, Encyclopedia Math. Appl., 13, Addison-Wesley Publishing Co., Reading, Mass., 1981, xx+325 pp. | MR | Zbl

[8] L. Baratchart, M. Yattselev, “Convergent interpolation to Cauchy integrals over analytic arcs with Jacobi-type weights”, Int. Math. Res. Not. IMRN, 2010:22 (2010), 4211–4275 | DOI | MR | Zbl

[9] L. Baratchart, H. Stahl, M. Yattselev, “Weighted extremal domains and best rational approximation”, Adv. Math., 229:1 (2012), 357–407 | DOI | MR | Zbl

[10] B. Beckermann, V. Kalyagin, A. C. Matos, F. Wielonsky, “Equilibrium problems for vector potentials with semidefinite interaction matrices and constrained masses”, Constr. Approx., 37:1 (2013), 101–134 | DOI | MR | Zbl

[11] V. I. Buslaev, S. P. Suetin, “Existence of compact sets with minimum capacity in problems of rational approximation of multivalued analytic functions”, Russian Math. Surveys, 69:1 (2014), 159–161 | DOI | DOI | MR | Zbl

[12] V. I. Buslaev, S. P. Suetin, “An extremal problem in potential theory”, Russian Math. Surveys, 69:5 (2014), 915–917 | DOI | DOI | MR | Zbl | Zbl

[13] V. I. Buslaev, “Convergence of $m$-point Padé approximants of a tuple of multivalued analytic functions”, Sb. Math., 206:2 (2015), 175–200 | DOI | DOI | MR | Zbl | Zbl

[14] V. I. Buslaev, “Capacity of a compact set in a logarithmic potential field”, Proc. Steklov Inst. Math., 290 (2015), 238–255 | DOI | DOI | MR | Zbl | Zbl

[15] V. I. Buslaev, S. P. Suetin, “On equilibrium problems related to the distribution of zeros of the Hermite–Padé polynomials”, Proc. Steklov Inst. Math., 290:1 (2015), 256–263 | DOI | DOI | MR | Zbl | Zbl

[16] V. I. Buslaev, S. P. Suetin, “On the existence of compacta of minimal capacity in the theory of rational approximation of multi-valued analytic functions”, J. Approx. Theory, 206 (2016), 48–67 | DOI | MR | Zbl

[17] P. Tchébycheff [Chebyshev], “Sur les fractions continues”, J. Math. Pures Appl. Sér. 2, 3 (1858), 289–323 | MR

[18] A. Deaño, D. Huybrechs, A. B. J. Kuijlaars, “Asymptotic zero distribution of complex orthogonal polynomials associated with Gaussian quadrature”, J. Approx. Theory, 162:12 (2010), 2202–2224 | DOI | MR | Zbl

[19] V. D. Erokhin, “O nailuchshem priblizhenii analiticheskikh funktsii posredstvom ratsionalnykh drobei so svobodnymi polyusami”, Dokl. AN SSSR, 128 (1959), 29–32 | MR | Zbl

[20] A. Hardy, A. B. J. Kuijlaars, “Weakly admissible vector equilibrium problems”, J. Approx. Theory, 170 (2013), 44–58 | DOI | MR | Zbl

[21] A. A. Gončar, “On the rapidity of rational approximation of continuous functions with characteristic singularities”, Math. USSR-Sb., 2:4 (1967), 561–568 | DOI | MR | Zbl

[22] A. A. Gonchar, “A local condition of single-valuedness of analytic functions”, Math. USSR-Sb., 18:1 (1972), 151–167 | DOI | MR | Zbl

[23] A. A. Gončar, “The rate of rational approximation and the property of single-valuedness of an analytic function in the neighborhood of an isolated singular point”, Math. USSR-Sb., 23:2 (1974), 254–270 | DOI | MR | Zbl

[24] A. A. Gončar, “A local condition for the single-valuedness of analytic functions of several variables”, Math. USSR-Sb., 22:2 (1974), 305–322 | DOI | MR | Zbl

[25] A. A. Gončar, “On the speed of rational approximation of some analytic functions”, Math. USSR-Sb., 34:2 (1978), 131–145 | DOI | MR | Zbl | Zbl

[26] A. A. Gonchar, “5.6. Rational approximation of analytic functions”, J. Soviet Math., 26:5 (1984), 2218–2220 | DOI | MR

[27] A. A. Gončar, G. López Lagomasino, “On Markov's theorem for multipoint Padé approximants”, Math. USSR-Sb., 34:4 (1978), 449–459 | DOI | MR | Zbl | Zbl

[28] A. A. Gonchar, E. A. Rakhmanov, “On convergence of simultaneous Padé approximants for systems of functions of Markov type”, Proc. Steklov Inst. Math., 157 (1983), 31–50 | MR | Zbl | Zbl

[29] A. A. Gonchar, “On the degree of rational approximation of analytic functions”, Proc. Steklov Inst. Math., 166 (1986), 53–61 | MR | Zbl | Zbl

[30] A. A. Gonchar, “Rational approximation of analytic functions”, Linear and complex analysis problem book, Lecture Notes in Math., 1043, Springer-Verlag, Berlin, 1984, 471–474 | DOI | MR | Zbl

[31] A. A. Gonchar, E. A. Rakhmanov, “Equilibrium measure and the distribution of zeros of extremal polynomials”, Math. USSR-Sb., 53:1 (1986), 119–130 | DOI | MR | Zbl

[32] A. A. Gonchar, E. A. Rakhmanov, “On the equilibrium problem for vector potentials”, Russian Math. Surveys, 40:4 (1985), 183–184 | DOI | MR | MR | Zbl

[33] A. A. Gonchar, “Rational approximations of analytic functions”, Nine papers from the International Congress of Mathematicians 1986, Amer. Math. Soc. Transl. Ser. 2, 147, Amer. Math. Soc., Providence, RI, 1990, 25–34 | DOI | MR | MR | Zbl

[34] A. A. Gonchar, E. A. Rakhmanov, “Equilibrium distributions and degree of rational approximation of analytic functions”, Math. USSR-Sb., 62:2 (1989), 305–348 | DOI | MR | Zbl | Zbl

[35] A. A. Gonchar, E. A. Rakhmanov, V. N. Sorokin, “Hermite–Padé approximants for systems of Markov-type functions”, Sb. Math., 188:5 (1997), 671–696 | DOI | DOI | MR | Zbl

[36] A. A. Gonchar, “Rational approximation of analytic functions”, Proc. Steklov Inst. Math., 272, suppl. 2 (2011), S44–S57 | DOI | DOI | MR | MR | Zbl | Zbl

[37] G. López Lagomasino, A. Martínez-Finkelshtein, P. Nevai, E. B. Saff, “Andrei Aleksandrovich Gonchar, November 21, 1931–October 10, 2012”, J. Approx. Theory, 172 (2013), A1–A13 | DOI | MR | Zbl

[38] A. V. Komlov, N. G. Kruzhilin, R. V. Palvelev, S. P. Suetin, “Convergence of Shafer quadratic approximants”, Russian Math. Surveys, 71:2 (2016), 373–375 | DOI | DOI | MR | Zbl

[39] A. B. J. Kuijlaars, G. L. F. Silva, “$S$-curves in polynomial external fields”, J. Approx. Theory, 191 (2015), 1–37 | DOI | MR | Zbl

[40] A. P. Magnus, “On the use of Carathéodory–Fejér method for investigating $\frac19$ and similar constants”, Nonlinear numerical methods and rational approximation (Wilrijk, 1987), Math. Appl., 43, Reidel, Dordrecht, 1988, 105–132 | MR | Zbl

[41] A. Markoff, “Deux démonstrations de la convergence de certaines fractions continues”, Acta Math., 19:1 (1895), 93–104 | DOI | MR | Zbl

[42] A. Martínez-Finkelshtein, E. Rakhmanov, “Critical measures, quadratic differentials, and weak limits of zeros of Stieltjes polynomials”, Comm. Math. Phys., 302:1 (2011), 53–111 | DOI | MR | Zbl

[43] A. Martínez-Finkelshtein, E. A. Rakhmanov, S. P. Suetin, “Heine, Hilbert, Padé, Riemann, and Stieltjes: John Nuttall's work 25 years later”, Recent advances in orthogonal polynomials, special functions, and their applications, Contemp. Math., 578, Amer. Math. Soc., Providence, RI, 2012, 165–193 | DOI | MR | Zbl

[44] S. N. Mergelyan, “O nekotorykh rezultatakh v teorii ravnomernykh i nailuchshikh priblizhenii polinomami i ratsionalnymi funktsiyami”, Prilozhenie k kn.: Dzh. L. Uolsh, Interpolyatsiya i approksimatsiya ratsionalnymi funktsiyami v kompleksnoi oblasti, IL, M., 1961, 461–499 | MR

[45] D. J. Newman, “Rational approximation to $|x|$”, Michigan Math. J., 11 (1964), 11–14 | DOI | MR | Zbl

[46] J. Nuttall, R. S. Singh, “Orthogonal polynomials and Padé approximants associated with a system of arcs”, J. Approximation Theory, 21:1 (1977), 1–42 | DOI | MR | Zbl

[47] J. Nuttall, “Asymptotics of diagonal Hermite–Padé polynomials”, J. Approx. Theory, 42:4 (1984), 299–386 | DOI | MR | Zbl

[48] O. G. Parfenov, “Estimates of the singular numbers of the Carleson imbedding operator”, Math. USSR-Sb., 59:2 (1988), 497–514 | DOI | MR | Zbl

[49] E. Perevoznikova, E. Rakhmanov, Variations of the equilibrium energy and ${S}$-property of compacta of minimal capacity, Manuscript (in Russian), 1994

[50] Ch. Pommerenke, “Padé approximants and convergence in capacity”, J. Math. Anal. Appl., 41:3 (1973), 775–780 | DOI | MR | Zbl

[51] V. A. Prokhorov, “On a theorem of Adamyan, Arov, and Kreĭn”, Russian Acad. Sci. Sb. Math., 78:1 (1994), 77–90 | DOI | MR | Zbl

[52] V. A. Prokhorov, “Rational approximation of analytic functions”, Russian Acad. Sci. Sb. Math., 78:1 (1994), 139–164 | DOI | MR | Zbl

[53] V. A. Prokhorov, “On the degree of rational approximation of meromorphic functions”, Russian Acad. Sci. Sb. Math., 81:1 (1995), 1–20 | DOI | MR | Zbl

[54] E. A. Rakhmanov, S. P. Suetin, “The distribution of the zeros of the Hermite–Padé polynomials for a pair of functions forming a Nikishin system”, Sb. Math., 204:9 (2013), 1347–1390 | DOI | DOI | MR | Zbl

[55] E. A. Rakhmanov, “Orthogonal polynomials and $S$-curves”, Recent advances in orthogonal polynomials, special functions, and their applications, Contemp. Math., 578, Amer. Math. Soc., Providence, RI, 2012, 195–239 | DOI | MR | Zbl

[56] H. Stahl, “Extremal domains associated with an analytic function. I”, Complex Variables Theory Appl., 4:4 (1985), 311–324 | DOI | MR | Zbl

[57] H. Stahl, “Extremal domains associated with an analytic function. II”, Complex Variables Theory Appl., 4:4 (1985), 325–338 | DOI | MR | Zbl

[58] H. Stahl, “The structure of extremal domains associated with an analytic function”, Complex Variables Theory Appl., 4:4 (1985), 339–354 | DOI | MR | Zbl

[59] H. Stahl, “Orthogonal polynomials with complex-valued weight function. I”, Constr. Approx., 2:3 (1986), 225–240 | DOI | MR | Zbl

[60] H. Stahl, “Orthogonal polynomials with complex-valued weight function. II”, Constr. Approx., 2:3 (1986), 241–251 | DOI | MR | Zbl

[61] H. Stahl, “Best uniform rational approximation of $|x|$ on $[-1,1]$”, Russian Acad. Sci. Sb. Math., 76:2 (1993), 461–487 | DOI | MR | Zbl

[62] H. Stahl, V. Totik, General orthogonal polynomials, Encyclopedia Math. Appl., 43, Cambridge Univ. Press, Cambridge, 1992, xii+250 pp. | DOI | MR | Zbl

[63] K. Strebel, Quadratic differentials, Ergeb. Math. Grenzgeb. (3), 5, Springer-Verlag, Berlin, 1984, xii+184 pp. | DOI | MR | Zbl

[64] S. P. Suetin, “Distribution of the zeros of Padé polynomials and analytic continuation”, Russian Math. Surveys, 70:5 (2015), 901–951 | DOI | DOI | MR | Zbl

[65] Ch.-J. De la Vallee Poussin, “Sur les polynomes d'approximation à une variable complexe”, Bull. Acad. Roy. de Belgique Cl. Sci., 3 (1911), 199–211

[66] R. S. Varga, A. Ruttan, A. J. Carpenter, “Numerical results on best uniform rational approximation of $|x|$ on $[-1,1]$”, Math. USSR-Sb., 74:2 (1993), 271–290 | DOI | MR | Zbl | Zbl

[67] N. S. Vyacheslavov, “Approximation of the function $|x|$ by rational functions”, Math. Notes, 16:1 (1974), 680–685 | DOI | MR | Zbl

[68] J. L. Walsh, “On approximation to an analytic function by rational functions of best approximation”, Math. Z., 38:1 (1934), 163–176 | DOI | MR | Zbl

[69] J. L. Walsh, Interpolation and approximation by rational functions in the complex domain, Amer. Math. Soc. Colloq. Publ., XX, 3rd ed., Amer. Math. Soc., Providence, RI, 1960, x+398 pp. | MR | MR | Zbl | Zbl

[70] J. L. Walsh, “Padé approximants as limits of rational functions of best approximation, real domain”, J. Approximation Theory, 11:3 (1974), 225–230 | DOI | MR | Zbl