Geodesic
Necessary and sufficient conditions for extending a function to a Schur function
Sbornik. Mathematics, Tome 211 (2020) no. 12, pp. 1660-1703 Cet article a éte moissonné depuis la source Math-Net.Ru

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A criterion for a function given by its values (with multiplicities) at a sequence of points in the disc $\mathbb D=\{|z|<1\}$ to extend to a holomorphic function in $\mathbb D$ with modulus at most $1$ is stated and proved. In the case when the function is defined by the values of its derivatives at $z=0$, this coincides with Schur's well-known criterion. Bibliography: 16 titles.
Keywords: continued fractions, Schur functions, Hankel determinants. 
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V. I. Buslaev. Necessary and sufficient conditions for extending a function to a Schur function. Sbornik. Mathematics, Tome 211 (2020) no. 12, pp. 1660-1703. http://geodesic.mathdoc.fr/item/SM_2020_211_12_a0/

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

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

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

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

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

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

[14] 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 | DOI | DOI | MR | Zbl

[15] V. I. Buslaev, “On the Van Vleck theorem for limit-periodic continued fractions of general form”, Proc. Steklov Inst. Math., 298 (2017), 68–93 | DOI | DOI | MR | Zbl

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