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Uspekhi Mat. Nauk, 2011, Volume 66, Issue 6(402), Pages 37–122 (Mi umn9448)  

This article is cited in 46 scientific papers (total in 46 papers)

Padé approximants, continued fractions, and orthogonal polynomials

A. I. Aptekareva, V. I. Buslaevb, A. Martínez-Finkelshteinc, S. P. Suetinb

a M. V. Keldysh Institute for Applied Mathematics, Russian Academy of Sciences, Moscow
b Steklov Mathematical Institute, Russian Academy of Sciences
c University of Almeria, Spain

Abstract: This is a survey of results constituting the foundations of the modern convergence theory of Padé approximants.
Bibliography: 204 titles.

Keywords: rational approximation, orthogonal polynomials, Padé approximants, equilibrium distributions, stationary compact sets, $S$-property.


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English version:
Russian Mathematical Surveys, 2011, 66:6, 1049–1131

Bibliographic databases:

UDC: 517.53
MSC: Primary 41A21; Secondary 30B70, 42C05
Received: 03.10.2011

Citation: A. I. Aptekarev, V. I. Buslaev, A. Martínez-Finkelshtein, S. P. Suetin, “Padé approximants, continued fractions, and orthogonal polynomials”, Uspekhi Mat. Nauk, 66:6(402) (2011), 37–122; Russian Math. Surveys, 66:6 (2011), 1049–1131

Citation in format AMSBIB
\by A.~I.~Aptekarev, V.~I.~Buslaev, A.~Mart{\'\i}nez-Finkelshtein, S.~P.~Suetin
\paper Pad\'e approximants, continued~fractions, and orthogonal polynomials
\jour Uspekhi Mat. Nauk
\yr 2011
\vol 66
\issue 6(402)
\pages 37--122
\jour Russian Math. Surveys
\yr 2011
\vol 66
\issue 6
\pages 1049--1131

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    This publication is cited in the following articles:
    1. 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
    2. A. V. Komlov, S. P. Suetin, “Widom's formula for the leading coefficient of a polynomial which is orthonormal with respect to a varying weight”, Russian Math. Surveys, 67:1 (2012), 183–185  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    3. M. A. Lapik, “Equilibrium measure for the vector logarithmic potential problem with an external field and the Nikishin interaction matrix”, Russian Math. Surveys, 67:3 (2012), 579–581  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    4. A. V. Komlov, S. P. Suetin, “An asymptotic formula for polynomials orthonormal with respect to a varying weight”, Trans. Moscow Math. Soc., 73 (2012), 139–159  mathnet  crossref  mathscinet  zmath  elib
    5. 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
    6. V. I. Buslaev, “Convergence of multipoint Padé approximants of piecewise analytic functions”, Sb. Math., 204:2 (2013), 190–222  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    7. A. P. Starovoitov, “Ermitovskaya approksimatsiya dvukh eksponent”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 13:1(2) (2013), 87–91  mathnet  crossref  elib
    8. A. P. Starovoitov, “Approksimatsii Ermita–Pade dlya sistemy funktsii Mittag-Lefflera”, PFMT, 2013, no. 1(14), 81–87  mathnet
    9. 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
    10. 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
    11. R. Jedynak, J. Gilewicz, “Computation of the $c$-table related to the Padé approximation”, Journal of Applied Mathematics, 2013 (2013), 185648, 10 pp.  crossref  mathscinet  isi
    12. J. Cacoq, B. de la Calle Ysern, G. Lopez Lagomasino, “and inverse results on row sequences of Hermite-Padé approximants”, Constr. Approx., 38:1 (2013), 133–160  crossref  mathscinet  zmath  isi  elib
    13. 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
    14. A. P. Starovoitov, “Asimptotika kvadratichnykh approksimatsii Ermita–Pade eksponentsialnykh funktsii”, PFMT, 2014, no. 1(18), 74–80  mathnet
    15. A. V. Komlov, S. P. Suetin, “Strong asymptotics of two-point Padé approximants for power-like multivalued functions”, Dokl. Math., 89:2 (2014), 1–4  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    16. E. A. Karabut, A. A. Kuzhuget, “Conformal mapping, Padé approximants, and an example of flow with a significant deformation of the free boundary”, European J. Appl. Math., 25:6 (2014), 729–747  crossref  mathscinet  zmath  isi  elib
    17. A. P. Starovoitov, “On asymptotic form of the Hermite–Pade approximations for a system of Mittag-Leffler functions”, Russian Math. (Iz. VUZ), 58:9 (2014), 49–56  mathnet  crossref
    18. M. A. Lapik, “Formula Buyarova–Rakhmanova dlya vneshnego polya v vektornoi zadache ravnovesiya logarifmicheskogo potentsiala”, Preprinty IPM im. M. V. Keldysha, 2014, 082, 15 pp.  mathnet
    19. M. A. Lapik, “Families of vector measures which are equilibrium measures in an external field”, Sb. Math., 206:2 (2015), 211–224  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    20. A. Martínez-Finkelshtein, Orive R., Rakhmanov E.A., “Phase transitions and equilibrium measures in random matrix models”, Comm. Math. Phys., 333:3 (2015), 1109–1173  crossref  mathscinet  zmath  isi
    21. A. V. Astafeva, “Asimptotika diagonalnykh approksimatsii Ermita–Pade dlya sistemy iz chetyrekh eksponent”, PFMT, 2015, no. 1(22), 53–57  mathnet
    22. A. P. Starovoitov, E. P. Kechko, “O lokalizatsii nulei approksimatsii Ermita–Pade eksponentsialnykh funktsii”, PFMT, 2015, no. 3(24), 84–89  mathnet
    23. 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
    24. N. R. Ikonomov, R. K. Kovacheva, S. P. Suetin, “Nuttall's integral equation and Bernshtein's asymptotic formula for a complex weight”, Izv. Math., 79:6 (2015), 1215–1234  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
    25. A. P. Starovoitov, E. P. Kechko, “Upper Bounds for the Moduli of Zeros of Hermite–Padé Approximations for a Set of Exponential Functions”, Math. Notes, 99:3 (2016), 417–425  mathnet  crossref  crossref  mathscinet  isi  elib
    26. A. V. Astafieva, A. P. Starovoitov, “Hermite-Padé approximation of exponential functions”, Sb. Math., 207:6 (2016), 769–791  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    27. 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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    28. A. I. Aptekarev, M. L. Yattselev, “Priblizheniya algebraicheskikh funktsii ratsionalnymi — funktsionalnye analogi diofantovykh approksimatsii”, Preprinty IPM im. M. V. Keldysha, 2016, 084, 24 pp.  mathnet  crossref
    29. Martinez-Finkelshtein A., Rakhmanov E.A., Suetin S.P., “Asymptotics of Type I Hermite–Padé Polynomials for Semiclassical Functions”, Modern Trends in Constructive Function Theory, Contemporary Mathematics, 661, eds. Hardin D., Lubinsky D., Simanek B., Amer Mathematical Soc, 2016, 199+  crossref  mathscinet  zmath  isi
    30. Song Han-Jie, Zhang Jin-Hai, Yao Zhen-Xing, “Normal moveout for long offset in isotropic media using the Padé approximation”, Appl. Geophys., 13:4 (2016), 658–666  crossref  isi  scopus
    31. Martinez-Finkelshtein A. Rakhmanov E.A., Do Orthogonal Polynomials Dream of Symmetric Curves?, Found. Comput. Math., 16:6 (2016), 1697–1736  crossref  mathscinet  zmath  isi  scopus
    32. Dyachenko S.A., Lushnikov P.M., Korotkevich A.O., “Branch Cuts of Stokes Wave on Deep Water. Part I: Numerical Solution and Padé Approximation”, Stud. Appl. Math., 137:4 (2016), 419–472  crossref  mathscinet  zmath  isi  elib  scopus
    33. Martinez-Finkelshtein A., Silva G.L.F., “Critical measures for vector energy: Global structure of trajectories of quadratic differentials”, Adv. Math., 302 (2016), 1137–1232  crossref  mathscinet  zmath  isi  elib  scopus
    34. Song H., Gao Y., Zhang J., Yao Zh., “Long-offset moveout for VTI using Padé approximation”, Geophysics, 81:5 (2016), C219–C227  crossref  isi  scopus
    35. M. V. Sidortsov, N. A. Starovoitova, A. P. Starovoitov, “Ob asimptotike approksimatsii Ermita–Pade vtorogo roda dlya eksponentsialnykh funktsii s kompleksnymi mnozhitelyami v pokazatelyakh eksponent”, PFMT, 2017, no. 1(30), 73–77  mathnet
    36. M. V. Sidortsov, A. A. Drapeza, A. P. Starovoitov, “Approksimatsii Ermita–Pade vyrozhdennykh gipergeometricheskikh funktsii”, PFMT, 2017, no. 2(31), 69–74  mathnet
    37. A. P. Starovoitov, “Asymptotics of Diagonal Hermite–Padé Polynomials for the Collection of Exponential Functions”, Math. Notes, 102:2 (2017), 277–288  mathnet  crossref  crossref  mathscinet  isi  elib
    38. 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  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    39. A. P. Starovoitov, E. P. Kechko, “On Some Properties of Hermite–Padé Approximants to an Exponential System”, Proc. Steklov Inst. Math., 298 (2017), 317–333  mathnet  crossref  crossref  isi  elib
    40. M. V. Sidortsov, A. A. Drapeza, A. P. Starovoitov, “Skorost skhodimosti kvadratichnykh approksimatsii Ermita–Pade vyrozhdennykh gipergeometricheskikh funktsii”, PFMT, 2018, no. 1(34), 71–78  mathnet
    41. 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
    42. A. P. Starovoitov, “Hermite–Padé approximants of the Mittag-Leffler functions”, Proc. Steklov Inst. Math., 301 (2018), 228–244  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    43. Karabut E.A., Petrov A.G., Zhuravleva E.N., “Semi-Analytical Study of the Voinovs Problem”, Eur. J. Appl. Math., 30:2 (2019), 298–337  crossref  mathscinet  isi  scopus
    44. Martinez-Finkelshtein A., Silva G.L.F., “Critical Measures For Vector Energy: Asymptotics of Non-Diagonal Multiple Orthogonal Polynomials For a Cubic Weight”, Adv. Math., 349 (2019), 246–315  crossref  isi
    45. S. P. Suetin, “Two Examples Related to Properties of Discrete Measures”, Math. Notes, 110:4 (2021), 578–582  mathnet  crossref  crossref  isi
    46. A. V. Komlov, “Polinomialnaya $m$-sistema Ermita–Pade dlya meromorfnykh funktsii na kompaktnoi rimanovoi poverkhnosti”, Matem. sb., 212:12 (2021), 40–76  mathnet  crossref
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