This article is cited in 23 scientific papers (total in 23 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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M. A. Lapik, “Families of vector measures which are equilibrium measures in an external field”, Sb. Math., 206:2 (2015), 211–224
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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
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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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