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Uspekhi Mat. Nauk, 2002, Volume 57, Issue 1(343), Pages 45–142 (Mi umn475)  

This article is cited in 59 scientific papers (total in 59 papers)

Padé approximants and efficient analytic continuation of a power series

S. P. Suetin

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: This survey reflects the current state of the theory of Padé approximants, that is, best rational approximations of power series. The main focus is on the so-called inverse problems of this theory, in which one must make deductions about analytic continuation of a given power series on the basis of the known asymptotic behaviour of the poles of some sequence of Padé approximants of this series. Row and diagonal sequences are studied from this point of view. Gonchar's and Rakhmanov's fundamental results of inverse nature are presented along with results of the author.

DOI: https://doi.org/10.4213/rm475

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English version:
Russian Mathematical Surveys, 2002, 57:1, 43–141

Bibliographic databases:

UDC: 517.53
MSC: Primary 41A21, 30B40, 41A20; Secondary 30E25, 30F30, 30B70, 34M50, 14K20
Received: 15.10.2001

Citation: S. P. Suetin, “Padé approximants and efficient analytic continuation of a power series”, Uspekhi Mat. Nauk, 57:1(343) (2002), 45–142; Russian Math. Surveys, 57:1 (2002), 43–141

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. S. P. Suetin, “On Dumas' theorem in the theory of continued fractions”, Russian Math. Surveys, 57:5 (2002), 1010–1012  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    2. S. P. Suetin, “Approximation properties of the poles of diagonal Padé approximants for certain generalizations of Markov functions”, Sb. Math., 193:12 (2002), 1837–1866  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    3. V. I. Buslaev, S. F. Buslaeva, “On the Rogers–Ramanujan Periodic Continued Fraction”, Math. Notes, 74:6 (2003), 783–793  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    4. S. P. Suetin, “The asymptotic behaviour of diagonal Padé approximants for hyperelliptic functions of genus $g=2$”, Russian Math. Surveys, 58:4 (2003), 802–804  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    5. S. P. Suetin, “Convergence of Chebyshëv continued fractions for elliptic functions”, Sb. Math., 194:12 (2003), 1807–1835  mathnet  crossref  crossref  mathscinet  zmath  isi
    6. S. L. Skorokhodov, “Padé approximation and numerical analysis for the Riemann $\zeta$-function”, Comput. Math. Math. Phys., 43:9 (2003), 1277–1298  mathnet  mathscinet  zmath
    7. Bogolyubsky A.I., Skorokhodov S.L., “Padé approximants, symbolic evaluations, and computation of solitons in two-field antiferromagnet model”, Program. Comput. Software, 30:2 (2004), 95–99  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    8. Celletti A., Falcolini C., Locatelli U., “On the break-down threshold of invariant tori in four dimensional maps”, Regul. Chaotic Dyn., 9:3 (2004), 227–253  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    9. A. L. Lukashov, “Circular parameters of polynomials orthogonal on several arcs of the unit circle”, Sb. Math., 195:11 (2004), 1639–1663  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    10. A. A. Gonchar, S. P. Suetin, “On Padé Approximants of Meromorphic Functions of Markov Type”, Proc. Steklov Inst. Math., 272, suppl. 2 (2011), S58–S95  mathnet  crossref  crossref  mathscinet  zmath  isi
    11. S. L. Skorokhodov, “Methods of analytical continuation of the generalized hypergeometric functions $ _pF_{p-1}(a_1,…,a_p;b_1,…,b_{p-1};z)$”, Comput. Math. Math. Phys., 44:7 (2004), 1102–1123  mathnet  mathscinet  zmath
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    13. Lukashov A.L., Peherstorfer F., “Zeros of polynomials orthogonal on two arcs of the unit circle”, J. Approx. Theory, 132:1 (2005), 42–71  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    14. S. L. Skorokhodov, “A method for computing the generalized hypergeometric function $ _pF_{p-1}(a_1,…,a_p;b_1,…,b_{p-1};1)$ in terms of the riemann zeta function”, Comput. Math. Math. Phys., 45:4 (2005), 550–562  mathnet  mathscinet  zmath  elib  elib
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    16. Ismail M.H., Matalgah M.M., “Performance of dual maximal ratio combining diversity in nonidentical correlated Weibull fading channels using Pade/spl acute/ approximation”, IEEE Trans. Commun., 54:3 (2006), 397–402  crossref  mathscinet  isi  elib  scopus  scopus
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    19. S. L. Skorokhodov, D. V. Khristoforov, “Calculation of the branch points of the eigenfunctions corresponding to wave spheroidal functions”, Comput. Math. Math. Phys., 46:7 (2006), 1132–1146  mathnet  crossref  mathscinet  elib
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    23. Ismail M.H., “Exact and approximate error-rate analysis of BPSK in Weibull fading with cochannel interference”, IET Commun., 1:2 (2007), 203–208  crossref  mathscinet  isi  elib  scopus  scopus
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    28. A. P. Starovoitov, N. A. Starovoitova, “On the Asymptotics of the Rows of the Padé Table of Analytic Functions with Logarithmic Branch Points”, Math. Notes, 84:3 (2008), 379–388  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
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    31. A. L. Lukashov, “Asymptotics of the averages of orthogonal polynomials on two intervals”, Siberian Math. J., 50:3 (2009), 474–480  mathnet  crossref  mathscinet  isi  elib
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