This article is cited in 22 scientific papers (total in 22 papers)
The distribution of the zeros of the Hermite-Padé polynomials for a pair of functions forming a Nikishin system
E. A. Rakhmanovab, S. P. Suetinb
a University of South Florida
b Steklov Mathematical Institute of the Russian Academy of Sciences
The distribution of the zeros of the Hermite-Padé polynomials of the first kind for a pair of functions with an arbitrary even number of common branch points lying on the real axis is investigated under the assumption
that this pair of functions forms a generalized complex Nikishin system. It is proved (Theorem 1) that the zeros have a limiting distribution, which coincides with the equilibrium measure of a certain compact set having the $\mathscr S$-property in a harmonic external field. The existence problem for $\mathscr S$-compact sets is solved in Theorem 2.
The main idea of the proof of Theorem 1 consists in replacing a vector equilibrium problem in potential theory by a scalar problem with an external field and then using the general Gonchar-Rakhmanov method, which was worked out in the solution of the ‘$1/9$’-conjecture.
The relation of the result obtained here to some results and conjectures due to Nuttall is discussed.
Bibliography: 51 titles.
orthogonal polynomials, Hermite-Padé polynomials, distribution of zeros, stationary compact set, Nuttall
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Sbornik: Mathematics, 2013, 204:9, 1347–1390
MSC: 26C10, 41A10, 42C05
Received: 29.08.2012 and 10.06.2013
E. A. Rakhmanov, S. P. Suetin, “The distribution of the zeros of the Hermite-Padé polynomials for a pair of functions forming a Nikishin system”, Mat. Sb., 204:9 (2013), 115–160; Sb. Math., 204:9 (2013), 1347–1390
Citation in format AMSBIB
\by E.~A.~Rakhmanov, S.~P.~Suetin
\paper The distribution of the zeros of the Hermite-Pad\'e polynomials for a~pair of functions forming a~Nikishin system
\jour Mat. Sb.
\jour Sb. Math.
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V. I. Buslaev, S. P. Suetin, “An extremal problem in potential theory”, Russian Math. Surveys, 69:5 (2014), 915–917
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A. B. J. Kuijlaars, G. L. F. Silva, “S-curves in polynomial external fields”, J. Approx. Theory, 191 (2015), 1–37
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V. I. Buslaev, S. P. Suetin, “On equilibrium problems related to the distribution of zeros of the Hermite–Padé polynomials”, Proc. Steklov Inst. Math., 290:1 (2015), 256–263
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A. V. Komlov, S. P. Suetin, “Distribution of the zeros of Hermite–Padé polynomials”, Russian Math. Surveys, 70:6 (2015), 1179–1181
M. A. Lapik, “Families of vector measures which are equilibrium measures in an external field”, Sb. Math., 206:2 (2015), 211–224
A. V. Komlov, N. G. Kruzhilin, R. V. Palvelev, S. P. Suetin, “Convergence of Shafer quadratic approximants”, Russian Math. Surveys, 71:2 (2016), 373–375
E. A. Rakhmanov, “The Gonchar-Stahl $\rho^2$-theorem and associated directions in the theory of rational approximations of analytic functions”, Sb. Math., 207:9 (2016), 1236–1266
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A. Martinez-Finkelshtein, E. A. Rakhmanov, S. P. Suetin, “Asymptotics of type I Hermite–Padé polynomials for semiclassical functions”, Modern trends in constructive function theory, Contemp. Math., 661, ed. D. Hardin, D. Lubinsky, B. Simanek, Amer. Math. Soc., Providence, RI, 2016, 199–228
S. P. Suetin, “On the distribution of the zeros of the Hermite–Padé polynomials for a quadruple of functions”, Russian Math. Surveys, 72:2 (2017), 375–377
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
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G. López Lagomasino, W. Van Assche, “Riemann-Hilbert analysis for a Nikishin system”, Sb. Math., 209:7 (2018), 1019–1050
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S. P. Suetin, “Equivalence of a Scalar and a Vector Equilibrium Problem for a Pair of Functions Forming a Nikishin System”, Math. Notes, 106:6 (2019), 971–980
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