General information
Latest issue
Impact factor
License agreement
Submit a manuscript

Search papers
Search references

Latest issue
Current issues
Archive issues
What is RSS

Uspekhi Mat. Nauk:

Personal entry:
Save password
Forgotten password?

Uspekhi Mat. Nauk, 2011, Volume 66, Issue 6(402), Pages 123–190 (Mi umn9454)  

This article is cited in 41 scientific papers (total in 42 papers)

Hermite–Padé approximations and multiple orthogonal polynomial ensembles

A. I. Aptekareva, A. Kuijlaarsb

a M. V. Keldysh Institute for Applied Mathematics, Russian Academy of Sciences, Moscow
b Katholieke Universiteit Leuven, Belgium

Abstract: This paper is concerned with Hermite–Padé rational approximants of analytic functions and their connection with multiple orthogonal polynomial ensembles of random matrices. Results on the analytic theory of such approximants are discussed, namely, convergence and the distribution of the poles of the rational approximants, and a survey is given of results on the distribution of the eigenvalues of the corresponding random matrices and on various regimes of such distributions. An important notion used to describe and to prove these kinds of results is the equilibrium of vector potentials with interaction matrices. This notion was introduced by A. A. Gonchar and E. A. Rakhmanov in 1981.
Bibliography: 91 titles.

Keywords: Hermite–Padé approximants, multiple orthogonal polynomials, weak and strong asymptotics, extremal equilibrium problems for a system of measures, matrix Riemann–Hilbert problem, Christoffel–Darboux formula, matrix model with an external source, non-intersecting paths, two-matrix model.


Full text: PDF file (1539 kB)
References: PDF file   HTML file

English version:
Russian Mathematical Surveys, 2011, 66:6, 1133–1199

Bibliographic databases:

UDC: 517.53
MSC: Primary 41A21, 42C05, 60B20; Secondary 31A15, 60G17, 60G55
Received: 15.09.2011

Citation: A. I. Aptekarev, A. Kuijlaars, “Hermite–Padé approximations and multiple orthogonal polynomial ensembles”, Uspekhi Mat. Nauk, 66:6(402) (2011), 123–190; Russian Math. Surveys, 66:6 (2011), 1133–1199

Citation in format AMSBIB
\by A.~I.~Aptekarev, A.~Kuijlaars
\paper Hermite--Pad\'e approximations and multiple orthogonal polynomial ensembles
\jour Uspekhi Mat. Nauk
\yr 2011
\vol 66
\issue 6(402)
\pages 123--190
\jour Russian Math. Surveys
\yr 2011
\vol 66
\issue 6
\pages 1133--1199

Linking options:

    SHARE: FaceBook Twitter Livejournal

    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. 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
    2. 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
    3. 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
    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. A. P. Starovoitov, G. N. Kazimirov, A. V. Astafeva, “Asimptotika approksimatsii Ermita-Pade dlya dvukh eksponent”, Vesnik Vitsebskaga dzyarzhainaga universiteta, 6:72 (2012), 23–27  elib
    6. A. Hardy, A. B. J. Kuijlaars, “Weakly admissible vector equilibrium problems”, J. Approx. Theory, 164:6 (2012), 854–868  crossref  mathscinet  zmath  isi  scopus
    7. S. Delvaux, A. López, G. López Lagomasino, “A family of Nikishin systems with periodic recurrence coefficients”, Sb. Math., 204:1 (2013), 43–74  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    8. 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
    9. A. P. Starovoitov, “Approksimatsii Ermita–Pade dlya sistemy funktsii Mittag-Lefflera”, PFMT, 2013, no. 1(14), 81–87  mathnet
    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. A. Hardy, A. B. J. Kuijlaars, “Weakly admissible vector equilibrium problems”, J. Approx. Theory, 170 (2013), 44–58  crossref  mathscinet  zmath  isi  scopus
    12. 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
    13. A. Aptekarev, J. Arvesu, “Asymptotics for multiple Meixner polynomials”, J. Math. Anal. Appl., 411:2 (2014), 485–505  crossref  mathscinet  zmath  isi  scopus
    14. 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
    15. A. I. Aptekarev, D. N. Tulyakov, “The leading term of the Plancherel-Rotach asymptotic formula for solutions of recurrence relations”, Sb. Math., 205:12 (2014), 1696–1719  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    16. 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
    17. A. I. Aptekarev, M. Derevyagin, W. Van Assche, “On 2D discrete Schrödinger operators associated with multiple orthogonal polynomials”, J. Phys. A, 48:6 (2015), 065201, 16 pp.  crossref  mathscinet  zmath  isi  scopus
    18. 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
    19. 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
    20. V. M. Buchstaber, V. N. Dubinin, V. A. Kaliaguine, B. S. Kashin, V. N. Sorokin, S. P. Suetin, D. N. Tulyakov, B. N. Chetverushkin, E. M. Chirka, A. A. Shkalikov, “Alexander Ivanovich Aptekarev (on his 60th birthday)”, Russian Math. Surveys, 70:5 (2015), 965–973  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    21. Aptekarev A.I., Lopez Lagomasino G., Martinez-Finkelshtein A., “Strong Asymptotics For the Pollaczek Multiple Orthogonal Polynomials”, 92, no. 3, 2015, 709–713  crossref  mathscinet  zmath  isi
    22. 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
    23. Emil Horozov, “Automorphisms of Algebras and Bochner's Property for Vector Orthogonal Polynomials”, SIGMA, 12 (2016), 050, 14 pp.  mathnet  crossref
    24. Arno B. J. Kuijlaars, “A Vector Equilibrium Problem for Muttalib–Borodin Biorthogonal Ensembles”, SIGMA, 12 (2016), 065, 15 pp.  mathnet  crossref
    25. V. G. Lysov, D. N. Tulyakov, “O vektornoi teoretiko-potentsialnoi zadache s matritsei Anzhelesko”, Preprinty IPM im. M. V. Keldysha, 2016, 110, 36 pp.  mathnet  crossref
    26. Aptekarev A.I., Derevyagin M., Van Assche W., “Discrete integrable systems generated by Hermite-Padé approximants”, Nonlinearity, 29:5 (2016), 1487–1506  crossref  mathscinet  zmath  isi  elib  scopus
    27. 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
    28. 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
    29. 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
    30. 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
    31. A. I. Aptekarev, G. López Lagomasino, A. Martínez-Finkelshtein, “On Nikishin systems with discrete components and weak asymptotics of multiple orthogonal polynomials”, Russian Math. Surveys, 72:3 (2017), 389–449  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    32. M. V. Sidortsov, A. A. Drapeza, A. P. Starovoitov, “Approksimatsii Ermita–Pade vyrozhdennykh gipergeometricheskikh funktsii”, PFMT, 2017, no. 2(31), 69–74  mathnet
    33. 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
    34. E. D. Belega, D. N. Tulyakov, “High-dimensional oscillators and Laguerre polynomials”, Russian Math. Surveys, 72:5 (2017), 965–967  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    35. V. G. Lysov, “Silnaya asimptotika approksimatsii Ermita–Pade dlya sistemy Nikishina s vesami Yakobi”, Preprinty IPM im. M. V. Keldysha, 2017, 085, 35 pp.  mathnet  crossref
    36. M. A. Lapik, D. N. Tulyakov, “Raspredelenie nulei mnogochlenov Ermita vblizi nulya i gaussovskie unitarnye ansambli”, Preprinty IPM im. M. V. Keldysha, 2017, 129, 11 pp.  mathnet  crossref
    37. Horozov E., “Vector Orthogonal Polynomials With Bochner'S Property”, Constr. Approx., 48:2 (2018), 201–234  crossref  mathscinet  isi  scopus
    38. 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
    39. Emil Horozov, “$d$-Orthogonal Analogs of Classical Orthogonal Polynomials”, SIGMA, 14 (2018), 063, 27 pp.  mathnet  crossref
    40. 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
    41. A. P. Starovoitov, “Hermite–Padé approximants of the Mittag-Leffler functions”, Proc. Steklov Inst. Math., 301 (2018), 228–244  mathnet  crossref  crossref  isi  elib  elib
    42. 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
  • Успехи математических наук Russian Mathematical Surveys
    Number of views:
    This page:699
    Full text:171
    First page:29

    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2019