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Uspekhi Mat. Nauk, 2011, Volume 66, Issue 6(402), Pages 3–36 (Mi umn9452)  

This article is cited in 14 scientific papers (total in 14 papers)

Padé–Chebyshev approximants of multivalued analytic functions, variation of equilibrium energy, and the $S$-property of stationary compact sets

A. A. Gonchara, E. A. Rakhmanovba, S. P. Suetina

a Steklov Mathematical Institute, Russian Academy of Sciences
b University of South Florida

Abstract: Padé–Chebyshev approximants are considered for multivalued analytic functions that are real-valued on the unit interval $[-1,1]$. The focus is mainly on non-linear Padé–Chebyshev approximants. For such rational approximations an analogue is found of Stahl's theorem on convergence in capacity of the Padé approximants in the maximal domain of holomorphy of the given function. The rate of convergence is characterized in terms of the stationary compact set for the mixed equilibrium problem of Green-logarithmic potentials.
Bibliography: 79 titles.

Keywords: rational approximation, Padé approximants, Chebyshev polynomials, non-linear Padé–Chebyshev approximants, stationary compact set, Stahl's theorem, convergence in capacity.


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English version:
Russian Mathematical Surveys, 2011, 66:6, 1015–1048

Bibliographic databases:

UDC: 517.53
MSC: Primary 30E10, 31A15, 41A21; Secondary 30F99, 40A15
Received: 08.11.2011

Citation: 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”, Uspekhi Mat. Nauk, 66:6(402) (2011), 3–36; Russian Math. Surveys, 66:6 (2011), 1015–1048

Citation in format AMSBIB
\by A.~A.~Gonchar, E.~A.~Rakhmanov, S.~P.~Suetin
\paper Pad\'e--Chebyshev approximants of multivalued analytic functions, variation of equilibrium energy, and the $S$-property of stationary compact sets
\jour Uspekhi Mat. Nauk
\yr 2011
\vol 66
\issue 6(402)
\pages 3--36
\jour Russian Math. Surveys
\yr 2011
\vol 66
\issue 6
\pages 1015--1048

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    This publication is cited in the following articles:
    1. 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
    2. 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
    3. 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
    4. 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
    5. 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
    6. 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
    7. 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
    8. V. Michel, G. M. Henkin, “Bishop-Runge approximations and inversion of a Riemann-Klein theorem”, Sb. Math., 206:2 (2015), 311–332  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    9. 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
    10. Buslaev V.I. Suetin S.P., “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  elib  scopus
    11. V. N. Sorokin, “Multipoint Hermite–Padé approximants for three beta functions”, Trans. Moscow Math. Soc., 2018, 117–134  mathnet  crossref  elib
    12. Andrianov I.V., Olevskyi V.I., Shapka I.V., Naumenko T.S., “Technique of Pade-Type Multidimensional Approximations Application For Solving Some Problems in Mathematical Physics”, AIP Conference Proceedings, 2025, ed. Todorov M., Amer Inst Physics, 2018, 040002-1  crossref  isi
    13. 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  mathnet  crossref  crossref  mathscinet  isi  elib
    14. V. N. Sorokin, “Hermite-Padé approximants to the Weyl function and its derivative for discrete measures”, Sb. Math., 211:10 (2020), 1486–1502  mathnet  crossref  crossref  mathscinet  isi  elib
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