Geodesic
An analogue of Polya's theorem for piecewise holomorphic functions
Sbornik. Mathematics, Tome 206 (2015) no. 12, pp. 1707-1721 Cet article a éte moissonné depuis la source Math-Net.Ru

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A well-known result due to Polya for a function given by its holomorphic germ at $z=\infty$ is extended to the case of a piecewise holomorphic function on an arbitrary compact set in $\overline{\mathbb C}$. This result is applied to the problem of the existence of compact sets that have the minimum transfinite diameter in the external field of the logarithmic potential of a negative unit charge among all compact sets such that a certain multivalued analytic function is single-valued and piecewise holomorphic on their complement. Bibliography: 13 titles.
Keywords: rational approximations, continued fractions, Hankel determinants
Mots-clés : Padé approximants. 
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V. I. Buslaev. An analogue of Polya's theorem for piecewise holomorphic functions. Sbornik. Mathematics, Tome 206 (2015) no. 12, pp. 1707-1721. http://geodesic.mathdoc.fr/item/SM_2015_206_12_a2/

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

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

[3] 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

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

[5] G. Polya, “Beitrag zur Verallgemeinerung des Verzerrungssatzes auf mehrfach zusammenhängende Gebiete. III”, Sitzungsber. Preuss. Akad. Wiss. Phys.-Math. Kl., 1929 (1929), 55–62 | Zbl

[6] 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 (to appear) ; 2015, arXiv: 1505.06120 | DOI

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

[8] A. Martínez-Finkelshtein, E. A. Rakhmanov, S. P. Suetin, “Variation of the equilibrium energy and the $S$-property of stationary compact sets”, Sb. Math., 202:12 (2011), 1831–1852 | DOI | DOI | MR | Zbl

[9] 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

[10] 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

[11] 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

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

[13] V. I. Buslaev, “Otsenka emkosti mnozhestva osobennostei funktsii, zadannykh svoim razlozheniem v nepreryvnuyu drob”, Anal. Math., 39:1 (2013), 1–27 | DOI | MR | Zbl