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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.

DOI: https://doi.org/10.4213/rm9454

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
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    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
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    11. A. Hardy, A. B. J. Kuijlaars, “Weakly admissible vector equilibrium problems”, J. Approx. Theory, 170 (2013), 44–58  crossref  mathscinet  zmath  isi  scopus
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    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
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    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
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    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
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    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
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    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
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    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
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    37. Horozov E., “Vector Orthogonal Polynomials With Bochner'S Property”, Constr. Approx., 48:2 (2018), 201–234  crossref  mathscinet  isi  scopus
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    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
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