The polynomial Hermite-Padé $m$-system for meromorphic functions on a compact Riemann surface

A. V. Komlov

Sbornik. Mathematics, Tome 212 (2021) no. 12, pp. 1694-1729

Résumé

Given a tuple of $m+1$ germs of arbitrary analytic functions at a fixed point, we introduce the polynomial Hermite-Padé $m$-system, which includes the Hermite-Padé polynomials of types I and II. In the generic case we find the weak asymptotics of the polynomials of the Hermite-Padé $m$-system constructed from the tuple of germs of functions $1, f_1,\dots,f_m$ that are meromorphic on an $(m+1)$-sheeted compact Riemann surface $\mathfrak R$. We show that if $f_j = f^j$ for some meromorphic function $f$ on $\mathfrak R$, then with the help of the ratios of polynomials of the Hermite-Padé $m$-system we recover the values of $f$ on all sheets of the Nuttall partition of $\mathfrak R$, apart from the last sheet. Bibliography: 18 titles.

Détail

Keywords: rational approximation, weak asymptotics, Riemann surface.
Mots-clés : Hermite-Padé polynomials

Bibliographie

[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 | DOI | DOI | MR | Zbl

[2] A. I. Aptekarev, D. N. Tulyakov, “Nuttall's Abelian integral on the Riemann surface of the cube root of a polynomial of degree 3”, Izv. Math., 80:6 (2016), 997–1034 | DOI | DOI | MR | Zbl

[3] E. M. Chirka, “Rimanovy poverkhnosti”, Lekts. kursy NOTs, 1, MIAN, M., 2006, 3–105 | DOI | Zbl

[4] E. M. Chirka, “On the $\bar\partial $-problem with $L^2$-estimates on a Riemann surface”, Proc. Steklov Inst. Math., 290:1 (2015), 264–276 | DOI | DOI | MR | Zbl

[5] E. M. Chirka, “Potentials on a compact Riemann surface”, Proc. Steklov Inst. Math., 301 (2018), 272–303 | DOI | DOI | MR | Zbl

[6] P. Henrici, “An algorithm for analytic continuation”, SIAM J. Numer. Anal., 3:1 (1966), 67–78 | DOI | MR | Zbl

[7] A. Komlov, “Polynomial Hermite–Padé $m$-system and reconstruction of the values of algebraic functions”, Extended abstracts Fall 2019, Trends Math., 12, Birkhäuser, Cham, 2021, 113–121 | DOI

[8] A. V. Komlov, R. V. Palvelev, S. P. Suetin, E. M. Chirka, “Hermite–Padé approximants for meromorphic functions on a compact Riemann surface”, Russian Math. Surveys, 72:4 (2017), 671–706 | DOI | DOI | MR | Zbl

[9] G. López Lagomasino, S. Medina Peralta, J. Szmigielski, “Mixed type Hermite–Padé approximation inspired by the Degasperis–Procesi equation”, Adv. Math., 349 (2019), 813–838 | DOI | MR | Zbl

[10] V. G. Lysov, “Mixed type Hermite–Padé approximants for a Nikishin system”, Proc. Steklov Inst. Math., 311 (2020), 199–213 | DOI | DOI | MR | Zbl

[11] J. Nuttall, “Hermite–Padé approximants to functions meromorphic on a Riemann surface”, J. Approx. Theory, 32:3 (1981), 233–240 | DOI | MR | Zbl

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

[13] V. V. Prasolov, Problems and theorems in linear algebra, Transl. Math. Monogr., 134, Amer. Math. Soc., Providence, RI, 1994, xviii+225 pp. | DOI | MR | Zbl

[14] E. A. Rakhmanov, “Zero distribution for Angelesco Hermite–Padé polynomials”, Russian Math. Surveys, 73:3 (2018), 457–518 | DOI | DOI | MR | Zbl

[15] T. Ransford, Potential theory in the complex plane, London Math. Soc. Stud. Texts, 28, Cambridge Univ. Press, Cambridge, 1995, x+232 pp. | DOI | MR | Zbl

[16] G. de Rham, Variétés différentiables. Formes, courants, formes harmoniques, Actualités Sci. Ind., 1222, Publ. Inst. Math. Univ. Nancago, III, Hermann et Cie, Paris, 1955, vii+196 pp. | MR | 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] S. P. Suetin, Hermite–Padé polynomials and analytic continuation: new approach and some results, arXiv: 1806.08735