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Mat. Sb., 2011, Volume 202, Number 12, Pages 113–136 (Mi msb7854)  

This article is cited in 25 scientific papers (total in 25 papers)

Variation of the equilibrium energy and the $S$-property of stationary compact sets

A. Martínez-Finkelshteina, E. A. Rakhmanovbc, S. P. Suetinb

a University of Almeria, Spain
b Steklov Mathematical Institute, Russian Academy of Sciences
c University of South Florida, USA

Abstract: This paper studies a variation of the equilibrium energy for a certain fairly general functional which appears naturally in the solution of many rational approximation problems of multi-valued analytic functions.
The main result of this work states that for the energy functional under consideration and a certain class of admissible compact sets, related to the function to be approximated, the corresponding stationary compact set is fully characterized by the so-called $S$-property.
Bibliography: 38 titles.

Keywords: rational approximation, orthogonal polynomials, Padé approximants, equilibrium distributions, stationary compact set, $S$-property.
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English version:
Sbornik: Mathematics, 2011, 202:12, 1831–1852

Bibliographic databases:

UDC: 517.53
MSC: 31A15
Received: 14.02.2011 and 29.09.2011

Citation: A. Martínez-Finkelshtein, E. A. Rakhmanov, S. P. Suetin, “Variation of the equilibrium energy and the $S$-property of stationary compact sets”, Mat. Sb., 202:12 (2011), 113–136; Sb. Math., 202:12 (2011), 1831–1852

Citation in format AMSBIB
\by A.~Mart{\'\i}nez-Finkelshtein, E.~A.~Rakhmanov, S.~P.~Suetin
\paper Variation of the equilibrium energy and the $S$-property of stationary compact sets
\jour Mat. Sb.
\yr 2011
\vol 202
\issue 12
\pages 113--136
\jour Sb. Math.
\yr 2011
\vol 202
\issue 12
\pages 1831--1852

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    This publication is cited in the following articles:
    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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    2. A. A. Gonchar, E. A. Rakhmanov, S. P. Suetin, “Padé–Chebyshev approximants of multivalued analytic functions, variation of equilibrium energy, and the $S$-property of stationary compact sets”, Russian Math. Surveys, 66:6 (2011), 1015–1048  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    3. A. Martínez-Finkelshtein, E. A. Rakhmanov, S. P. Suetin, “Heine, Hilbert, Padé, Riemann, and Stieltjes: John Nuttall's work 25 years later”, Recent advances in orthogonal polynomials, special functions, and their applications, Contemp. Math., 578, ed. Arvesu J. Lagomasino G., Amer Mathematical Soc, 2011, 165–193  crossref  mathscinet  isi
    4. S. P. Suetin, “An analogue of the Hadamard and Schiffer variational formulas”, Theoret. and Math. Phys., 170:3 (2012), 274–279  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
    5. E. A. Rakhmanov, S. P. Suetin, “Asymptotic behaviour of the Hermite–Padé polynomials of the 1st kind for a pair of functions forming a Nikishin system”, Russian Math. Surveys, 67:5 (2012), 954–956  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    6. A. I. Aptekarev, D. N. Tulyakov, “Asymptotics of Meixner polynomials and Christoffel-Darboux kernels”, Trans. Moscow Math. Soc., 73 (2012), 67–106  mathnet  crossref  mathscinet  zmath  elib
    7. V. I. Buslaev, A. Martínez-Finkelshtein, S. P. Suetin, “Method of interior variations and existence of $S$-compact sets”, Proc. Steklov Inst. Math., 279 (2012), 25–51  mathnet  crossref  mathscinet  isi  elib
    8. A. V. Komlov, S. P. Suetin, “An asymptotic formula for a two-point analogue of Jacobi polynomials”, Russian Math. Surveys, 68:4 (2013), 779–781  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    9. 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”, Sb. Math., 204:9 (2013), 1347–1390  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    10. V. I. Buslaev, S. P. Suetin, “Existence of compact sets with minimum capacity in problems of rational approximation of multivalued analytic functions”, Russian Math. Surveys, 69:1 (2014), 159–161  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    11. R. K. Kovacheva, S. P. Suetin, “Distribution of zeros of the Hermite–Padé polynomials for a system of three functions, and the Nuttall condenser”, Proc. Steklov Inst. Math., 284 (2014), 168–191  mathnet  crossref  crossref  isi  elib  elib
    12. V. I. Buslaev, S. P. Suetin, “An extremal problem in potential theory”, Russian Math. Surveys, 69:5 (2014), 915–917  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    13. D. Huybrechs, A. B. J. Kuijlaars, N. Lejon, “Zero distribution of complex orthogonal polynomials with respect to exponential weights”, J. Approx. Theory, 184 (2014), 28–54  crossref  mathscinet  zmath  isi  elib  scopus
    14. A. B. J. Kuijlaars, G. L. F. Silva, “S-curves in polynomial external fields”, J. Approx. Theory, 191 (2015), 1–37  crossref  mathscinet  zmath  isi  elib  scopus
    15. 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  mathnet  crossref  crossref  isi  elib  elib
    16. V. I. Buslaev, “Capacity of a compact set in a logarithmic potential field”, Proc. Steklov Inst. Math., 290:1 (2015), 238–255  mathnet  crossref  crossref  isi  elib  elib
    17. S. P. Suetin, “Distribution of the zeros of Padé polynomials and analytic continuation”, Russian Math. Surveys, 70:5 (2015), 901–951  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    18. V. I. Buslaev, “An analogue of Polya's theorem for piecewise holomorphic functions”, Sb. Math., 206:12 (2015), 1707–1721  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    19. V. N. Sorokin, “Ob asimptoticheskikh rezhimakh sovmestnykh mnogochlenov Meiksnera”, Preprinty IPM im. M. V. Keldysha, 2016, 046, 32 pp.  mathnet  crossref
    20. V. I. Buslaev, S. P. Suetin, “On the existence of compacta of minimal capacity in the theory of rational approximation of multi-valued analytic functions”, J. Approx. Theory, 206:SI (2016), 48–67  crossref  mathscinet  zmath  isi  scopus
    21. V. N. Sorokin, “On Multiple Orthogonal Polynomials for Three Meixner Measures”, Proc. Steklov Inst. Math., 298 (2017), 294–316  mathnet  crossref  crossref  isi  elib
    22. E. A. Rakhmanov, “Zero distribution for Angelesco Hermite–Padé polynomials”, Russian Math. Surveys, 73:3 (2018), 457–518  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    23. M. A. Lapik, D. N. Tulyakov, “On expanding neighborhoods of local universality of Gaussian unitary ensembles”, Proc. Steklov Inst. Math., 301 (2018), 170–179  mathnet  crossref  crossref  isi  elib  elib
    24. V. N. Sorokin, “Multipoint Hermite–Padé approximants for three beta functions”, Trans. Moscow Math. Soc., 2018, 117–134  mathnet  crossref  elib
    25. V. N. Sorokin, “Approksimatsii Ermita–Pade funktsii Veilya i ee proizvodnoi dlya diskretnykh mer”, Matem. sb., 211:10 (2020), 139–156  mathnet  crossref
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