Abstract:
This paper considers the zero distribution of Hermite–Padé polynomials of the first kind associated with a vector function
→f=(f1,…,fs)
whose components fk are functions with a finite number of branch points in the plane. The branch sets of component functions are assumed to be sufficiently well separated (which constitutes the Angelesco case). Under this condition, a theorem on the limit zero
distribution for such polynomials is proved. The limit measures are defined in terms of a known vector equilibrium problem.
The proof of the theorem is based on methods developed by Stahl [59]–[63] and Gonchar and the author [27], [55].
These methods are generalized further in the paper in application to collections of polynomials defined by systems of complex orthogonality relations.
Together with the characterization of the limit zero distributions of Hermite–Padé polynomials in terms of a vector equilibrium problem, the paper considers an alternative characterization using a Riemann surface R(→f) associated with →f.
In these terms, a more general conjecture (without the Angelesco condition) on the zero distribution of Hermite–Padé polynomials is presented.
Bibliography: 72 titles.
\Bibitem{Rak18}
\by E.~A.~Rakhmanov
\paper Zero distribution for Angelesco Hermite--Pad\'e polynomials
\jour Russian Math. Surveys
\yr 2018
\vol 73
\issue 3
\pages 457--518
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This publication is cited in the following 17 articles:
S. P. Suetin, “O skalyarnykh podkhodakh k izucheniyu predelnogo raspredeleniya nulei mnogochlenov Ermita–Pade dlya sistemy Nikishina”, UMN, 80:1(481) (2025), 85–152
A. V. Komlov, R. V. Palvelev, “Zeros of discriminants constructed from Hermite–Padé polynomials of an algebraic function and their relation to branch points”, Sb. Math., 215:12 (2024), 1633–1665
A. P. Magnus, J. Meinguet, “Strong asymptotics of the best rational approximation to the exponential function on a bounded interval”, Sb. Math., 215:12 (2024), 1666–1719
S. P. Suetin, “Convergence of Hermite–Padé rational approximations”, Russian Math. Surveys, 78:5 (2023), 967–969
E. A. Rakhmanov, S. P. Suetin, “Approksimatsii Chebysheva–Pade dlya mnogoznachnykh funktsii”, Tr. MMO, 83, no. 2, MTsNMO, M., 2022, 319–344
V. G. Lysov, “Mnogourovnevye interpolyatsii dlya obobschennoi sistemy Nikishina na grafe-dereve”, Tr. MMO, 83, no. 2, MTsNMO, M., 2022, 345–361
N. R. Ikonomov, S. P. Suetin, “Struktura nattollovskogo razbieniya dlya nekotorogo klassa chetyrekhlistnykh rimanovykh poverkhnostei”, Tr. MMO, 83, no. 1, MTsNMO, M., 2022, 37–61
E. A. Rakhmanov, S. P. Suetin, “Chebyshev–Padé approximants for multivalued functions”, Trans. Moscow Math. Soc., –
V. G. Lysov, “Multilevel interpolations for the generalized Nikishin system on a tree graph”, Trans. Moscow Math. Soc., –
N. R. Ikonomov, S. P. Suetin, “Structure of the Nuttall partition for some class of four-sheeted Riemann surfaces”, Trans. Moscow Math. Soc., 2022, –
A. V. Komlov, “The polynomial Hermite-Padé m-system for meromorphic functions on a compact Riemann surface”, Sb. Math., 212:12 (2021), 1694–1729
I. A. Aptekarev, S. A. Denisov, M. L. Yattselev, “Self-adjoint Jacobi matrices on trees and multiple orthogonal polynomials”, Trans. Amer. Math. Soc., 373:2 (2020), 875–917
N. R. Ikonomov, S. P. Suetin, “Scalar Equilibrium Problem and the Limit Distribution of Zeros of Hermite–Padé Polynomials of Type II”, Proc. Steklov Inst. Math., 309 (2020), 159–182
S. P. Suetin, “Existence of a three-sheeted Nutall surface for a certain class of infinite-valued analytic functions”, Russian Math. Surveys, 74:2 (2019), 363–365
S.-Y. Lee, M. Yang, “Planar orthogonal polynomials as Type II multiple orthogonal polynomials”, J. Phys. A, 52:27 (2019), 275202, 14 pp.
S. P. Suetin, “On a new approach to the problem of distribution of zeros of Hermite–Padé polynomials for a Nikishin system”, Proc. Steklov Inst. Math., 301 (2018), 245–261
S. P. Suetin, “On an Example of the Nikishin System”, Math. Notes, 104:6 (2018), 905–914
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