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Schur’s criterion for formal power series. (English. Russian original) Zbl 1461.30007

Sb. Math. 210, No. 11, 1563-1580 (2019); translation from Mat. Sb. 210, No. 11, 58-75 (2019).
Summary: A criterion for when a formal power series can be represented by a formal Schur continued fraction is stated. The proof proposed is based on a relationship, revealed here, between Hankel two-point determinants of a series and its Schur determinants.

MSC:

30B10 Power series (including lacunary series) in one complex variable
30B70 Continued fractions; complex-analytic aspects
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References:

[1] W. B. Jones, O. Njåstad and W. J. Thron 1986 Schur fractions, Perron-Carathéodory fractions and Szegö polynomials, a survey Analytic theory of continued fractions IIPitlochry/Aviemore 1985 Lecture Notes in Math. 1199 Springer, Berlin 127-158 · Zbl 0596.30009 · doi:10.1007/BFb0075938
[2] J. Schur 1917 Über Potenzreihen, die im Innern des Einheitskreises beschränkt sind J. Reine Angew. Math.1917 147 205-232 · JFM 46.0475.01 · doi:10.1515/crll.1917.147.205
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[6] English transl. in V. I. Buslaev 2001 Proc. Steklov Inst. Math.235 29-43 · Zbl 1011.30004
[7] V. I. Buslaev 2010 On Hankel determinants of functions given by their expansions in \(P\)-fractions Ukrain. Math. Zh.62 3 315-326 · Zbl 1224.30012
[8] English transl. in V. I. Buslaev 2010 Ukrainian Math. J.62 3 358-372 · Zbl 1224.30012 · doi:10.1007/s11253-010-0359-x
[9] V. I. Buslaev 2013 An estimate for the capacity of singular sets of functions that are defined by their continued fraction expansions Anal. Math.39 1 1-27 · Zbl 1289.30017 · doi:10.1007/s10476-013-0101-7
[10] V. I. Buslaev 2016 An analog of Gonchar’s theorem for the \(m\)-point version of Leighton’s conjecture Function spaces, approximation theory and related areas of complex analysis Tr. Mat. Inst. Steklova 293 MAIK “Nauka/Interperiodika”, Moscow 133-145 · Zbl 1367.30004 · doi:10.1134/S0371968516020096
[11] English transl. in V. I. Buslaev 2016 Proc. Steklov Inst. Math.293 127-139 · Zbl 1367.30004 · doi:10.1134/S008154381604009X
[12] V. I. Buslaev 2017 On the Van Vleck theorem for limit-periodic continued fractions of general form Complex analysis and its applications Tr. Mat. Inst. Steklova 298 MAIK “Nauka/Interperiodika”, Moscow 75-100 · Zbl 1394.30001 · doi:10.1134/S0371968517030062
[13] English transl. in V. I. Buslaev 2017 Proc. Steklov Inst. Math.298 68-93 · Zbl 1394.30001 · doi:10.1134/S0081543817060062
[14] V. I. Buslaev 2018 On singular points of meromorphic functions determined by continued fractions Mat. Zametki103 4 490-502 · Zbl 1400.30004 · doi:10.4213/mzm11737
[15] English transl. in V. I. Buslaev 2018 Math. Notes103 4 527-536 · Zbl 1400.30004 · doi:10.1134/S0001434618030203
[16] V. I. Buslaev 2018 Continued fractions with limit periodic coefficients Mat. Sb.209 2 47-65 · Zbl 1393.30005 · doi:10.4213/sm8687
[17] English transl. in V. I. Buslaev 2018 Sb. Math.209 2 187-205 · Zbl 1393.30005 · doi:10.1070/SM8687
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