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Uspekhi Mat. Nauk, 2015, Volume 70, Issue 5(425), Pages 121–174 (Mi umn9675)  

This article is cited in 33 scientific papers (total in 33 papers)

Distribution of the zeros of Padé polynomials and analytic continuation

S. P. Suetin

Steklov Mathematical Institute of Russian Academy of Sciences

Abstract: The problem of analytic continuation of a multivalued analytic function with finitely many branch points on the Riemann sphere is discussed. The focus is on Padé approximants: classical (one-point) Padé approximants, multipoint Padé approximants, and Hermite–Padé approximants. The main result is a theorem on the distribution of zeros and the convergence of the Hermite–Padé approximants for a system $[1,f,f^2]$, where $f$ is a multivalued function in the so-called Laguerre class $\mathscr{L}$.
Bibliography: 128 titles.

Keywords: analytic continuation, continued fractions, orthogonal polynomials, rational approximants, Padé polynomials, Hermite–Padé polynomials, distribution of zeros, GRS-method, convergence in capacity.

Funding Agency Grant Number
Russian Science Foundation 14-50-00005
This work is supported by the Russian Science Foundation under grant 14-50-00005.


DOI: https://doi.org/10.4213/rm9675

Full text: PDF file (1191 kB)
References: PDF file   HTML file

English version:
Russian Mathematical Surveys, 2015, 70:5, 901–951

Bibliographic databases:

UDC: 517.53
MSC: Primary 30E10, 30B40, 41A20, 41A21; Secondary 12H05, 31A15
Received: 30.06.2015

Citation: S. P. Suetin, “Distribution of the zeros of Padé polynomials and analytic continuation”, Uspekhi Mat. Nauk, 70:5(425) (2015), 121–174; Russian Math. Surveys, 70:5 (2015), 901–951

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    This publication is cited in the following articles:
    1. A. V. Komlov, N. G. Kruzhilin, R. V. Palvelev, S. P. Suetin, “Convergence of Shafer quadratic approximants”, Russian Math. Surveys, 71:2 (2016), 373–375  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    2. 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
    3. V. I. Buslaev, “An analog of Gonchar's theorem for the $m$-point version of Leighton's conjecture”, Proc. Steklov Inst. Math., 293 (2016), 127–139  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    4. 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
    5. S. P. Suetin, “Zero distribution of Hermite–Padé polynomials and localization of branch points of multivalued analytic functions”, Russian Math. Surveys, 71:5 (2016), 976–978  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    6. V. I. Buslaev, “The Capacity of the Rational Preimage of a Compact Set”, Math. Notes, 100:6 (2016), 781–790  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    7. 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
    8. S. P. Suetin, “On the distribution of the zeros of the Hermite–Padé polynomials for a quadruple of functions”, Russian Math. Surveys, 72:2 (2017), 375–377  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    9. 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
    10. S. P. Suetin, “An Analog of Pólya's Theorem for Multivalued Analytic Functions with Finitely Many Branch Points”, Math. Notes, 101:5 (2017), 888–898  mathnet  crossref  crossref  mathscinet  isi  elib
    11. M. V. Sidortsov, A. A. Drapeza, A. P. Starovoitov, “Approksimatsii Ermita–Pade vyrozhdennykh gipergeometricheskikh funktsii”, PFMT, 2017, no. 2(31), 69–74  mathnet
    12. 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
    13. 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
    14. 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
    15. V. I. Buslaev, “On the Van Vleck Theorem for Limit-Periodic Continued Fractions of General Form”, Proc. Steklov Inst. Math., 298 (2017), 68–93  mathnet  crossref  crossref  mathscinet  isi  elib
    16. V. G. Lysov, D. N. Tulyakov, “On a Vector Potential-Theory Equilibrium Problem with the Angelesco Matrix”, Proc. Steklov Inst. Math., 298 (2017), 170–200  mathnet  crossref  crossref  isi  elib
    17. 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
    18. E. A. Karabut, A. G. Petrov, E. N. Zhuravleva, “Semi-analytical investigation of unsteady free-boundary flows”, Application of Mathematics in Technical and Natural Sciences, AIP Conf. Proc., 1895, no. 1, Amer. Inst. Phys., 2017, 120006  crossref  isi  scopus
    19. V. I. Buslaev, “Continued fractions with limit periodic coefficients”, Sb. Math., 209:2 (2018), 187–205  mathnet  crossref  crossref  adsnasa  isi  elib
    20. V. I. Buslaev, “On Singular points of Meromorphic Functions Determined by Continued Fractions”, Math. Notes, 103:4 (2018), 527–536  mathnet  crossref  crossref  isi  elib
    21. A. P. Starovoitov, M. V. Sidortsov, A. A. Drapeza, “Skorost skhodimosti kvadratichnykh approksimatsii Ermita–Pade vyrozhdennykh gipergeometricheskikh funktsii”, PFMT, 2018, no. 1(34), 71–78  mathnet
    22. S. P. Suetin, “Distribution of the zeros of Hermite–Padé polynomials for a complex Nikishin system”, Russian Math. Surveys, 73:2 (2018), 363–365  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    23. 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
    24. A. I. Aptekarev, E. D. Belega, “Asymptotic behaviour of the information entropy of hydrogen-like quantum systems”, Russian Math. Surveys, 73:5 (2018), 922–924  mathnet  crossref  crossref  adsnasa  isi  elib
    25. D. S. Lubinsky, “Exact interpolation, spurious poles, and uniform convergence of multipoint Padé approximants”, Sb. Math., 209:3 (2018), 432–448  mathnet  crossref  crossref  adsnasa  isi  elib
    26. S. P. Suetin, “On a new approach to the problem of distribution of zeros of Hermite–Padé polynomials for a Nikishin system”, Proc. Steklov Inst. Math., 301 (2018), 245–261  mathnet  crossref  crossref  isi  elib  elib
    27. V. G. Lysov, D. N. Tulyakov, “On the supports of vector equilibrium measures in the Angelesco problem with nested intervals”, Proc. Steklov Inst. Math., 301 (2018), 180–196  mathnet  crossref  crossref  isi  elib  elib
    28. 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
    29. E. M. Chirka, “Potentials on a compact Riemann surface”, Proc. Steklov Inst. Math., 301 (2018), 272–303  mathnet  crossref  crossref  isi  elib  elib
    30. S. I. Bezrodnykh, “The Lauricella hypergeometric function $F_D^{(N)}$, the Riemann–Hilbert problem, and some applications”, Russian Math. Surveys, 73:6 (2018), 941–1031  mathnet  crossref  crossref  adsnasa  isi  elib
    31. Lubinsky D.S., “On Uniform Convergence of Diagonal Multipoint Pade Approximants For Entire Functions”, Constr. Approx., 49:1 (2019), 149–174  crossref  mathscinet  zmath  isi
    32. 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
    33. Gopal A., Trefethen L.N., “Representation of Conformal Maps By Rational Functions”, Numer. Math., 142:2 (2019), 359–382  crossref  mathscinet  zmath  isi  scopus
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