Geodesic
On a~lower bound for the rate of convergence of multipoint Pad\' e approximants of piecewise analytic functions
Izvestiya. Mathematics , Tome 85 (2021) no. 3, pp. 351-366.

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We obtain a lower bound for the rate of convergence of multipoint Padé approximants of functions holomorphically extendable from a compact set to a union of domains whose boundaries possess a symmetry property. The bound obtained matches a known upper bound for the same quantity.
Keywords: rational approximation, convergence in capacity.
Mots-clés : orthogonal polynomials, Padé approximants, distribution of poles 
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V. I. Buslaev. On a~lower bound for the rate of convergence of multipoint Pad\' e approximants of piecewise analytic functions. Izvestiya. Mathematics , Tome 85 (2021) no. 3, pp. 351-366. http://geodesic.mathdoc.fr/item/IM2_2021_85_3_a2/

[1] 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

[2] V. I. Buslaev, “Convergence of multipoint Padé approximants of piecewise analytic functions”, Sb. Math., 204:2 (2013), 190–222 | DOI | DOI | MR | Zbl

[3] E. B. Saff, V. Totik, Logarithmic potentials with external fields, Grundlehren Math. Wiss., 316, Springer-Verlag, Berlin, 1997, xvi+505 pp. | DOI | MR | Zbl

[4] H. Stahl, “Orthogonal polynomials with complex-valued weight function. I”, Constr. Approx., 2:3 (1986), 225–240 ; II, 241–251 | DOI | MR | Zbl | Zbl

[5] H. Stahl, “The convergence of Padé approximants to functions with branch points”, J. Approx. Theory, 91:2 (1997), 139–204 | DOI | MR | Zbl

[6] 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 ; 2011, arXiv: 1112.5713 | DOI | MR | Zbl

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

[8] V. I. Buslaev, A. Martínez-Finkelshtein, S. P. Suetin, “Method of interior variations and existence of $S$-compact sets”, Proc. Steklov Inst. Math., 279 (2012), 25–51 | DOI | MR | Zbl

[9] G. V. Kuz'mina, “Moduli of families of curves and quadratic differentials”, Proc. Steklov Inst. Math., 139 (1982), 1–231 | MR | MR | Zbl | Zbl

[10] 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

[11] 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

[12] V. I. Buslaev, “Capacity of a compact set in a logarithmic potential field”, Proc. Steklov Inst. Math., 290:1 (2015), 238–255 | DOI | DOI | MR | Zbl

[13] 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

[14] V. I. Buslaev, “On the convergence of continued T-fractions”, Proc. Steklov Inst. Math., 235 (2001), 29–43 | MR | Zbl

[15] V. I. Buslaev, “On singular points of meromorphic functions determined by continued fractions”, Math. Notes, 103:4 (2018), 527–536 | DOI | DOI | MR | Zbl

[16] V. I. Buslaev, “Continued fractions with limit periodic coefficients”, Sb. Math., 209:2 (2018), 187–205 | DOI | DOI | MR | Zbl

[17] V. I. Buslaev, “Schur's criterion for formal power series”, Sb. Math., 210:11 (2019), 1563–1580 | DOI | DOI | MR | Zbl

[18] V. I. Buslaev, “Convergence of a limit periodic Schur continued fraction”, Math. Notes, 107:5 (2020), 701–712 | DOI | DOI | MR | Zbl

[19] V. I. Buslaev, “On the Van Vleck theorem for regular $C$-fractions with limit-periodic coefficients”, Izv. Math., 65:4 (2001), 673–686 | DOI | DOI | MR | Zbl

[20] V. I. Buslaev, “An analogue of Polya's theorem for piecewise holomorphic functions”, Sb. Math., 206:12 (2015), 1707–1721 | DOI | DOI | MR | Zbl

[21] V. I. Buslaev, “The capacity of the rational preimage of a compact set”, Math. Notes, 100:6 (2016), 781–789 | DOI | DOI | MR | Zbl

[22] 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