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Trudy Mat. Inst. Steklova, 2001, Volume 235, Pages 36–51 (Mi tm232)  

This article is cited in 13 scientific papers (total in 13 papers)

On the Convergence of Continued T-Fractions

V. I. Buslaev

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: It is shown that a continued $\mathrm T$-fraction converges on the set $\{|z|<R_1\} \cup \{|z|>R_2\}$. Formulas (exact in a certain sense) for evaluating the radii $R_1$ and $R_2$ of these disks are given. For a $\mathrm T$-fraction with limit-periodic coefficients, a cut $\Gamma$ on the complex plane is explicitly specified such that this $\mathrm T$-fraction converges outside this cut. It is shown that the meromorphic function represented by this $\mathrm T$-fraction cannot be meromorphically continued (as a single-valued function) across any arc lying on $\Gamma$.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2001, 235, 29–43

Bibliographic databases:
UDC: 517.55
Received in March 2001

Citation: V. I. Buslaev, “On the Convergence of Continued T-Fractions”, Analytic and geometric issues of complex analysis, Collected papers. Dedicated to the 70th anniversary of academician Anatolii Georgievich Vitushkin, Trudy Mat. Inst. Steklova, 235, Nauka, MAIK Nauka/Inteperiodika, M., 2001, 36–51; Proc. Steklov Inst. Math., 235 (2001), 29–43

Citation in format AMSBIB
\by V.~I.~Buslaev
\paper On the Convergence of Continued T-Fractions
\inbook Analytic and geometric issues of complex analysis
\bookinfo Collected papers. Dedicated to the 70th anniversary of academician Anatolii Georgievich Vitushkin
\serial Trudy Mat. Inst. Steklova
\yr 2001
\vol 235
\pages 36--51
\publ Nauka, MAIK Nauka/Inteperiodika
\publaddr M.
\jour Proc. Steklov Inst. Math.
\yr 2001
\vol 235
\pages 29--43

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    This publication is cited in the following articles:
    1. 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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    2. 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  mathnet  crossref  mathscinet  isi  elib
    3. V. I. Buslaev, “Convergence of multipoint Padé approximants of piecewise analytic functions”, Sb. Math., 204:2 (2013), 190–222  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    4. Buslaev V.I., “An Estimate of the Capacity of Singular Sets of Functions That Are Defined by Continued Fractions”, Anal. Math., 39:1 (2013), 1–27  crossref  mathscinet  zmath  isi  elib  scopus
    5. S. P. Suetin, “Distribution of the zeros of Padé polynomials and analytic continuation”, Russian Math. Surveys, 70:5 (2015), 901–951  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    6. V. I. Buslaev, “An analogue of Polya's theorem for piecewise holomorphic functions”, Sb. Math., 206:12 (2015), 1707–1721  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    7. Buslaev V.I. Suetin S.P., “On the existence of compacta of minimal capacity in the theory of rational approximation of multi-valued analytic functions”, J. Approx. Theory, 206:SI (2016), 48–67  crossref  mathscinet  zmath  isi  elib  scopus
    8. V. I. Buslaev, “On the Van Vleck Theorem for Limit-Periodic Continued Fractions of General Form”, Proc. Steklov Inst. Math., 298 (2017), 68–93  mathnet  crossref  crossref  mathscinet  isi  elib
    9. V. I. Buslaev, “Continued fractions with limit periodic coefficients”, Sb. Math., 209:2 (2018), 187–205  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    10. V. I. Buslaev, “Schur's criterion for formal power series”, Sb. Math., 210:11 (2019), 1563–1580  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    11. V. I. Buslaev, “Convergence of a Limit Periodic Schur Continued Fraction”, Math. Notes, 107:5 (2020), 701–712  mathnet  crossref  crossref  mathscinet  isi  elib
    12. V. I. Buslaev, “Necessary and sufficient conditions for extending a function to a Schur function”, Sb. Math., 211:12 (2020), 1660–1703  mathnet  crossref  crossref  mathscinet  isi  elib
    13. 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  mathnet  crossref  crossref  isi  elib
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