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 Mat. Sb., 2015, Volume 206, Number 12, Pages 55–69 (Mi msb8557)

An analogue of Polya's theorem for piecewise holomorphic functions

V. I. Buslaev

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

Abstract: 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, Padé approximants.

 Funding Agency Grant Number Russian Science Foundation 14-50-00005 This work is supported by the Russian Science Foundation under grant no. 14-50-00005.

DOI: https://doi.org/10.4213/sm8557

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English version:
Sbornik: Mathematics, 2015, 206:12, 1707–1721

Bibliographic databases:

UDC: 517.53
MSC: 30C80, 31A15

Citation: V. I. Buslaev, “An analogue of Polya's theorem for piecewise holomorphic functions”, Mat. Sb., 206:12 (2015), 55–69; Sb. Math., 206:12 (2015), 1707–1721

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. I. Buslaev, “An analog of Gonchar's theorem for the $m$-point version of Leighton's conjecture”, Proc. Steklov Inst. Math., 293 (2016), 127–139
2. V. I. Buslaev, “The Capacity of the Rational Preimage of a Compact Set”, Math. Notes, 100:6 (2016), 781–790
3. S. P. Suetin, “An Analog of Pólya's Theorem for Multivalued Analytic Functions with Finitely Many Branch Points”, Math. Notes, 101:5 (2017), 888–898
4. V. I. Buslaev, “On the Van Vleck Theorem for Limit-Periodic Continued Fractions of General Form”, Proc. Steklov Inst. Math., 298 (2017), 68–93
5. V. I. Buslaev, “Continued fractions with limit periodic coefficients”, Sb. Math., 209:2 (2018), 187–205
6. V. I. Buslaev, “On Singular points of Meromorphic Functions Determined by Continued Fractions”, Math. Notes, 103:4 (2018), 527–536
7. E. A. Rakhmanov, “Zero distribution for Angelesco Hermite–Padé polynomials”, Russian Math. Surveys, 73:3 (2018), 457–518
8. 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
9. V. I. Buslaev, “Convergence of a Limit Periodic Schur Continued Fraction”, Math. Notes, 107:5 (2020), 701–712
10. V. I. Buslaev, “Schur's Criterion for Formal Newton Series”, Math. Notes, 108:6 (2020), 884–888
11. V. I. Buslaev, “Necessary and sufficient conditions for extending a function to a Schur function”, Sb. Math., 211:12 (2020), 1660–1703
12. V. I. Buslaev, “On a lower bound for the rate of convergence of multipoint Padé approximants of piecewise analytic functions”, Izv. Math., 85:3 (2021), 351–366
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