Geodesic
A direct proof of Stahl's theorem for a generic class of algebraic functions
Sbornik. Mathematics, Tome 213 (2022) no. 11, pp. 1582-1596 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Under the assumption that Stahl's $S$-compact set exists we give a short proof of the limiting distribution of the zeros of Padé polynomials and the convergence in capacity of diagonal Padé approximants for a generic class of algebraic functions. The proof is direct, rather than by contradiction as Stahl's original proof was. The ‘generic class’ means, in particular, that all the ramification points of the multisheeted Riemann surface of the algebraic function in question are of the second order (that is, all branch points of the function are of square root type). As a consequence, a conjecture of Gonchar relating to Padé approximations is proved for this class of algebraic functions. We do not use the relations of orthogonality for Padé polynomials. The proof is based on the maximum principle only. Bibliography: 19 titles.
Keywords: convergence in capacity, Stahl's theorem, Riemann surface.
Mots-clés : Padé approximant 
@article{SM_2022_213_11_a6,
     author = {S. P. Suetin},
     title = {A~direct proof of {Stahl's} theorem for a~generic class of algebraic functions},
     journal = {Sbornik. Mathematics},
     pages = {1582--1596},
     year = {2022},
     volume = {213},
     number = {11},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2022_213_11_a6/}
}
TY  - JOUR
AU  - S. P. Suetin
TI  - A direct proof of Stahl's theorem for a generic class of algebraic functions
JO  - Sbornik. Mathematics
PY  - 2022
SP  - 1582
EP  - 1596
VL  - 213
IS  - 11
UR  - http://geodesic.mathdoc.fr/item/SM_2022_213_11_a6/
LA  - en
ID  - SM_2022_213_11_a6
ER  - 
%0 Journal Article
%A S. P. Suetin
%T A direct proof of Stahl's theorem for a generic class of algebraic functions
%J Sbornik. Mathematics
%D 2022
%P 1582-1596
%V 213
%N 11
%U http://geodesic.mathdoc.fr/item/SM_2022_213_11_a6/
%G en
%F SM_2022_213_11_a6
S. P. Suetin. A direct proof of Stahl's theorem for a generic class of algebraic functions. Sbornik. Mathematics, Tome 213 (2022) no. 11, pp. 1582-1596. http://geodesic.mathdoc.fr/item/SM_2022_213_11_a6/

[1] A. I. Aptekarev, V. I. Buslaev, A. Martínez-Finkelshtein and S. P. Suetin, “Padé approximants, continued fractions, and orthogonal polynomials”, Uspekhi Mat. Nauk, 66:6(402) (2011), 37–122 ; English transl. in Russian Math. Surveys, 66:6 (2011), 1049–1131 | DOI | MR | Zbl | DOI

[2] A. I. Aptekarev and M. L. Yattselev, “Padé approximants for functions with branch points – strong asymptotics of Nuttall-Stahl polynomials”, Acta Math., 215:2 (2015), 217–280 | DOI | MR | Zbl

[3] V. I. Buslaev, “On a lower bound for the rate of convergence of multipoint Padé approximants of piecewise analytic functions”, Izv. Ross. Akad. Nauk Ser. Mat., 85:3 (2021), 13–29 ; English transl. in Izv. Math., 85:3 (2021), 351–366 | DOI | MR | Zbl | DOI

[4] E. M. Chirka, “Capacities on a compact Riemann surface”, Tr. Mat. Inst. Steklova, 311 (2020), 41–83 ; English transl. in Proc. Steklov Inst. Math., 311 (2020), 36–77 | DOI | MR | Zbl | DOI

[5] A. A. Gonchar and E. A. Rakhmanov, “Equilibrium distributions and degree of rational approximation of analytic functions”, Mat. Sb., 134(176):3(11) (1987), 306–352 ; English transl. in Sb. Math., 62:2 (1989), 305–348 | MR | Zbl | DOI

[6] A. V. Komlov, “The polynomial Hermite-Padé $m$-system for meromorphic functions on a compact Riemann surface”, Mat. Sb., 212:12 (2021), 40–76 ; English transl. in Sb. Math., 212:12 (2021), 1694–1729 | DOI | MR | Zbl | DOI

[7] N. S. Landkof, Foundations of modern potential theory, Nauka, Moscow, 1966, 515 pp. ; English transl., Grundlehren Math. Wiss., 180, Springer-Verlag, New York–Heidelberg, 1972, x+424 pp. | MR | Zbl | MR | Zbl

[8] A. Martínez-Finkelshtein, E. A. Rakhmanov and S. P. Suetin, “Variation of the equilibrium energy and the $S$-property of stationary compact sets”, Mt. Sb., 202:12 (2011), 113–136 ; English transl. in Sb. Math., 202:12 (2011), 1831–1852 | DOI | MR | Zbl | DOI

[9] J. Nuttall and S. R. Singh, “Orthogonal polynomials and Padé approximants associated with a system of arcs”, J. Approx. Theory, 21:1 (1977), 1–42 | DOI | MR | Zbl

[10] J. Nuttall, “Asymptotics of diagonal Hermite-Padé polynomials”, J. Approx. Theory, 42:4 (1984), 299–386 | DOI | MR | Zbl

[11] E. A. Perevoznikova and E. A. Rakhmanov, Variation of equilibrium energy and the $S$-property of compact sets with minimum capacity, Manuscript, 1994 (Russian)

[12] E. A. Rahmanov (Rakhmanov), “Convergence of diagonal Padé approximants”, Mat. Sb., 104(146):2(10) (1977), 271–291 ; English transl. in Sb. Math., 33:2 (1977), 243–260 | MR | Zbl | DOI

[13] E. A. Rakhmanov, “Orthogonal polynomials and $S$-curves”, Recent advances in orthogonal polynomials, special functions, and their applications, Contemp. Math., 578, Amer. Math. Soc., Providence, RI, 2012, 195–239 | DOI | MR | Zbl

[14] E. B. Saff and V. Totik, Logarithmic potentials with external fields, Appendix B by T. Bloom, Grundlehren Math. Wiss., 316, Springer-Verlag, Berlin, 1997, xvi+505 pp. | DOI | MR | Zbl

[15] H. Stahl, “Three different approaches to a proof of convergence for Padé approximants”, Rational approximation and applications in mathematics and physics (Łańcut 1985), Lecture Notes in Math., 1237, Springer, Berlin, 1987, 79–124 | DOI | MR | Zbl

[16] H. Stahl, “Diagonal Padé approximants to hyperelliptic functions”, Ann. Fac. Sci. Toulouse Math. (6), 1996, special issue, 121–193 | DOI | MR | Zbl

[17] H. Stahl, “The convergence of Padé approximants to functions with branch points”, J. Approx. Theory, 91:2 (1997), 139–204 | DOI | MR | Zbl

[18] H. R. Stahl, Sets of minimal capacity and extremal domains, arXiv: 1205.3811

[19] M. L. Yattselev, “Convergence of two-point Padé approximants to piecewise holomorphic functions”, Mat. Sb., 212:11 (2021), 128–164 ; English transl. in Sb. Math., 212:11 (2021), 1626–1659 | DOI | MR | Zbl | DOI