Abstract
We prove the equivalence of a vector and a scalar equilibrium problem that naturally arise when studying the limit distribution of zeros of type I Hermite–Padé polynomials for a pair of functions forming a Nikishin system.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. P. Suetin, “On a new approach to the problem of distribution of zeros of Hermite–Padépolynomials for a Nikishin system,” in Tr. Mat. Inst. Steklova, Vol. 301: Complex Analysis, Mathematical Physics, and Applications, Collected Papers (MAIK Nauka/Interperiodica, Moscow, 2018), pp. 259–275 [Proc. Steklov Inst. Math. 301, 245–261 (2018)].
S. P. Suetin, “Distribution of the zeros of Hermite–Padépolynomials for a complex Nikishin system,” UspekhiMat. Nauk 73 (2 (440)), 183–184 (2018) [RussianMath. Surveys 73 (2), 363–365 (2018)].
H. Stahl, “Three different approaches to a proof of convergence for Padéapproximants,” in Lecture Notes in Math., Vol. 1237: Rational Approximation and Applications in Mathematics and Physics (Springer-Verlag, Berlin, 1987), pp. 79–124.
H. Stahl, “Asymptotics of Hermite–Padépolynomials and related convergence results. A summary of results,” in Math. Appl., Vol. 43: Nonlinear Numerical Methods and Rational Approximation (Reidel, Dordrecht, 1988), pp. 23–53.
E. M. Nikishin, “Asymptotic behavior of linear forms for simultaneous Padéapproximants,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 2, 33–41 (1986) [SovietMath. (Iz. VUZ) 30 (2), 43–52 (1986)].
E.M. Nikishin and V. N. Sorokin, Rational Approximations and Orthogonality (Nauka, Moscow, 1988; Amer.Math. Soc., Providence, 1991).
A. I. Aptekarev, A. I. Bogolyubskii, and M. L. Yattselev, “Convergence of ray sequences of Frobenius–Padé approximants,” Mat. Sb. 208 (3), 4–27 (2017) [Sb.Math. 208 (3), 313–334 (2017)].
D. Barrios Rolańia, J. S. Geronimo, and G. López Lagomasino, “High-order recurrence relations, Hermite–Padéapproximation and Nikishin systems,” Mat. Sb. 209 (3), 102–137 (2018) [Sb. Math. 209 (3), 385–420 (2018)].
G. López Lagomasino and W. Van Assche, “Riemann-Hilbert analysis for a Nikishin system,” Mat. Sb. 209 (7), 106–138 (2018) [Sb.Math. 209 (7), 1019–1050 (2018)].
S. P. Suetin, “On an example of the Nikishin system,” Mat. Zametki 104 (6), 918–929 (2018) [Math. Notes 104 (6), 905–914 (2018)].
E. A. Rakhmanov and S. P. Suetin, “The distribution of the zeros of the Hermite-Pade polynomials for a pair of functions forming a Nikishin system,” Mat. Sb. [Sb. Math.] 204 (9), 115–160 (2013) [Sb. Math. 204 (9), 1347–1390 (2013)].
A. Martínez-Finkelshtein, E. A. Rakhmanov, and S. P. Suetin, “Asymptotics of type I Hermite–Padé polynomials for semiclassical functions,” in Contemp. Math., Vol. 661: Modern Trends in Constructive Function Theory (Amer.Math. Soc., Providence, RI, 2016), pp. 199–228.
S. P. Suetin, Hermite–Padé Polynomials and Analytic Continuation: New Approach and Some Results, arXiv:1806.08735 (2018).
J. Nuttall, “Asymptotics of diagonal Hermite–Padépolynomials,” J. Approx. Theory 42 (4), 299–386 (1984).
A. A. Gonchar and E. A. Rakhmanov, “On the convergence of simultaneous Padéapproximants for systems of functions of Markov type,” in Trudy Mat. Inst. Steklova, Vol. 157: Number theory, Mathematical Analysis, and Their Applications, Collection of Articles. Dedicated to I. M. Vinogradov, a member of the Academy of Sciences on the occasion of his 90th birthday (MIAN, Moscow, 1981), pp. 31–48 [Proc. Steklov Inst.Math. 157, 31–50 (1983)].
A. I. Aptekarev, “Asymptotics of Hermite–Padéapproximants for two functions with branch points,” Dokl. Ross. Akad. Nauk 422 (4), 443–445 (2008) [Dokl.Math. 78 (2), 717–719 (2008)].
E. A. Rakhmanov, “Orthogonal polynomials and S-curves,” in Contemp. Math., Vol. 578: Recent Advances in Orthogonal Polynomials, Special Functions and Their Applications (Amer. Math. Soc., Providence, RI, 2012), pp. 195–239.
V. I. Buslaev and S. P. Suetin, “On equilibrium problems related to the distribution of zeros of the Hermite–Padépolynomials,” in Trudy Mat. Inst. Steklova, Vol. 290: Modern Problems of Mathematics, Mechanics, and Mathematical Physics, Collected Papers (MAIK Nauka/Interperiodica, Moscow, 2015), pp. 272–279 [Proc. Steklov Inst. Math. 290 (1), 256–263 (2015)].
E.M. Chirka, “Potentials on a compact Riemann surface,” in TrudyMat. Inst. Steklova, Vol. 301: Complex Analysis, Mathematical Physics, and Applications, Collected Papers (MAIK Nauka/Interperiodica, Moscow, 2018), pp. 287–319 [Proc. Steklov Inst. Math. 301, 272–303 (2018)].
E. M. Chirka, “Equilibrium measures on a compact Riemann surface,” in Trudy Mat. Inst. Steklova, Vol. 306: Mathematical Physics, and Applications, Collected Papers (MAIK Nauka/Interperiodica, Moscow, 2019) (in press).
H. R. Stahl, Sets of Minimal Capacity and Extremal Domains, arXiv:1205.3811 (2012).
L. Baratchart, H. Stahl, and M. Yattselev, “Weighted extremal domains and best rational approximation,” Adv. Math. 229 (1), 357–407 (2012).
Funding
This work was supported by the Russian Science Foundation under grant 19-11-00316.
Author information
Authors and Affiliations
Corresponding author
Additional information
Russian Text © The Author(s), 2019, published in Matematicheskie Zametki, 2019, Vol. 106, No. 6, pp. 904–916.
Rights and permissions
About this article
Cite this article
Suetin, S.P. Equivalence of a Scalar and a Vector Equilibrium Problem for a Pair of Functions Forming a Nikishin System. Math Notes 106, 970–979 (2019). https://doi.org/10.1134/S0001434619110336
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434619110336