Skip to main content
SpringerLink
Log in

Convergence of a Limit Periodic Schur Continued Fraction

  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

In this paper, we show that if the parameters of a Schur continued fraction tend to zero, then the functions to which the even convergents converge inside the unit disk and the functions to which the odd convergents converge outside the unit disk cannot have a meromorphic continuation to each other through any arc of the unit circle. This result is obtained as a consequence of the convergence theorem for limit periodic Schur continued fractions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. I. Schur, “Über Potenzreihen, die im Innern des Einheitskreises beschränkt sind,” J. Reine Angew. Math. 147, 205–232 (1917).

    MathSciNet  MATH  Google Scholar 

  2. H.-J. Runkel, “Bounded analytic functions in the unit disk and the behaviour of certain analytic continued fractions near the singular line,” J. Reine Angew. Math. 281, 97–125 (1976).

    MathSciNet  Google Scholar 

  3. W. B. Jones, O. Njåstad, and W. J. Thron, “Schur fractions, Perron–Carathéodory fractions and Szegő polynomials, a survey,” in Lecture Notes in Math., Vol. 1199: Analytic Theory of Continued Fractions. II (Springer-Verlag, Berlin, 1986), pp. 127–158.

    Google Scholar 

  4. V. I. Buslaev, “Schur’s criterion for formal power series,” Mat. Sb. 210 (11), 58–75 (2019) [Sb. Math. 210 (11), 1563–1580 (2019)].

    Article  MathSciNet  Google Scholar 

  5. E. B. Saff and V. Totik, Logarithmic Potentials with External Fields, in Grundlehren Math. Wiss. (Springer-Verlag, Berlin, 1997), Vol. 316.

    Book  Google Scholar 

  6. V. I. Buslaev, “On the Van Vleck theorem for regular C-fractions with limit-periodic coefficients,” Izv. Ross. Akad. Nauk Ser. Mat. 65 (4), 35–48 (2001) [Izv. Math. 65 (4), 673–686 (2001)].

    Article  MathSciNet  Google Scholar 

  7. V. I. Buslaev, “On the convergence of continued T-fractions,” in Tr.Mat. Inst. Steklov., Vol. 235: Analytic and Geometric Problems of Complex Analysis (Nauka, MAIK “Nauka/Interperiodika”, Moscow, 2001), pp. 36–51 [Proc. Steklov Inst. Math. 235, 29–43 (2001)].

    Google Scholar 

  8. V. I. Buslaev, “The capacity of the rational preimage of a compact set,” Mat. Zametki 100 (6), 790–799 (2016) [Math. Notes 100 (6), 781–789 (2016)].

    Article  MathSciNet  Google Scholar 

  9. V. I. Buslaev, “An analogue of Polya’s theorem for piecewise holomorphic functions,” Mat. Sb. 206 (12), 55–69 (2015) [Sb. Math. 206 (12), 1707–1721 (2015)].

    Article  Google Scholar 

  10. G. Polya, “Sitzungsberichte Akad. Berlin,” (1929), pp. 55–62.

  11. V. I. Buslaev, “On Hankel determinants of functions given by their expansions in P-fractions,” Ukrain. Mat. Zh. 62 (3), 315–326 (2010) [Ukrainian Math. J. 62 (3), 358–372 (2010)].

    Article  MathSciNet  Google Scholar 

  12. V. I. Buslaev, “On singular points of meromorphic functions determined by continued fractions,” Mat. Zametki 103 (4), 490–502 (2018) [Math. Notes 103 (4), 527–536 (2018)].

    Article  MathSciNet  Google Scholar 

  13. V. I. Buslaev, “Continued fractions with limit periodic coefficients,” Mat. Sb. 209 (2), 47–65 (2018) [Sb. Math. 209 (2), 187–205 (2018)].

    Article  MathSciNet  Google Scholar 

Download references

Funding

This work was supported by the Russian Science Foundation under grant 19-11-00316.

Author information

Authors and Affiliations

  1. Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, 119991, Russia

    V. I. Buslaev

Authors
  1. V. I. Buslaev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. I. Buslaev.

Additional information

Russian Text © The Author(s), 2020, published in Matematicheskie Zametki, 2020, Vol. 107, No. 5, pp. 643–656.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Buslaev, V.I. Convergence of a Limit Periodic Schur Continued Fraction. Math Notes 107, 701–712 (2020). https://doi.org/10.1134/S0001434620050016

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0001434620050016

Keywords

Search

Navigation