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Mat. Sb. (N.S.), 1979, Volume 109(151), Number 4(8), Pages 629–646 (Mi msb2413)  

This article is cited in 15 scientific papers (total in 15 papers)

Inverse theorems on generalized Padé approximants

S. P. Suetin

Abstract: In this paper the following theorem is proved.
Theorem. {\it For $m>0$ and all sufficiently large $n$, let the Padé approximants $R_{n,m}$ of the series
$$ f(z)=\sum_{\nu=0}^\infty A_\nu F_\nu(z),\qquad A_\nu=(f,F_\nu)=\int_{-1}^1f(x)F_\nu(x) d\alpha(x), $$
have exactly $m$ finite poles, and let there exist a polynomial $\omega_m(z)=\prod_{j=1}^m(z-z_j)$ such that
$$ \varlimsup_{n\to\infty}\|q_{n,m}-\omega_m\|^{1/n}\leqslant\delta<1. $$
$$ \rho_m(f)\geqslant\frac1\delta\max_{1\leqslant j\leqslant m}|\varphi(z_j)| $$
and in the region $D_m(f)=D_{\rho_m}$ the function $f$ has exactly $m$ poles (at the points $z_1,…,z_m$). }
Bibliography: 8 titles.

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English version:
Mathematics of the USSR-Sbornik, 1980, 37:4, 581–597

Bibliographic databases:

UDC: 517.53
MSC: 30E10, 30D30
Received: 20.10.1978

Citation: S. P. Suetin, “Inverse theorems on generalized Padé approximants”, Mat. Sb. (N.S.), 109(151):4(8) (1979), 629–646; Math. USSR-Sb., 37:4 (1980), 581–597

Citation in format AMSBIB
\by S.~P.~Suetin
\paper Inverse theorems on generalized Pad\'e approximants
\jour Mat. Sb. (N.S.)
\yr 1979
\vol 109(151)
\issue 4(8)
\pages 629--646
\jour Math. USSR-Sb.
\yr 1980
\vol 37
\issue 4
\pages 581--597

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    This publication is cited in the following articles:
    1. V. V. Vavilov, G. L. Lopes, V. A. Prokhorov, “On an inverse problem for the rows of a Padé table”, Math. USSR-Sb., 38:1 (1981), 109–118  mathnet  crossref  mathscinet  zmath  isi
    2. Suetin S., “On Montessusdeballore Theorem for Non-Linear Pade Approximations of Orthogonal Expansions and Faber Series”, 253, no. 6, 1980, 1322–1325  mathscinet  zmath  isi
    3. A. A. Gonchar, “Poles of rows of the Padé table and meromorphic continuation of functions”, Math. USSR-Sb., 43:4 (1982), 527–546  mathnet  crossref  mathscinet  zmath
    4. Lagomasino G. Vavilov V., “Survey on Recent Advances in Inverse Problems of Pade-Approximation Theory”, 1105, 1984, 11–26  mathscinet  zmath  isi
    5. D. Barrios Rolanı́a, G. López Lagomasino, E.B. Saff, “Asymptotics of orthogonal polynomials inside the unit circle and Szegő–Padé approximants”, Journal of Computational and Applied Mathematics, 133:1-2 (2001), 171  crossref  mathscinet  zmath
    6. S. P. Suetin, “Padé approximants and efficient analytic continuation of a power series”, Russian Math. Surveys, 57:1 (2002), 43–141  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    7. D.Barrios Rolanı́a, G.López Lagomasino, E.B. Saff, “Determining radii of meromorphy via orthogonal polynomials on the unit circle”, Journal of Approximation Theory, 124:2 (2003), 263  crossref  mathscinet  zmath
    8. V. I. Buslaev, “On the Fabry Ratio Theorem for Orthogonal Series”, Proc. Steklov Inst. Math., 253 (2006), 8–21  mathnet  crossref  mathscinet  elib
    9. V. I. Buslaev, “Analog of the Hadamard Formula for the First Ellipse of Meromorphy”, Math. Notes, 85:4 (2009), 528–543  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    10. V. I. Buslaev, “An analogue of Fabry's theorem for generalized Padé approximants”, Sb. Math., 200:7 (2009), 981–1050  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    11. A. I. Aptekarev, V. I. Buslaev, A. Martínez-Finkelshtein, S. P. Suetin, “Padé approximants, continued fractions, and orthogonal polynomials”, Russian Math. Surveys, 66:6 (2011), 1049–1131  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    12. N. Bosuwan, G. López Lagomasino, “Inverse Theorem on Row Sequences of Linear Padé-orthogonal Approximation”, Comput. Methods Funct. Theory, 2015  crossref  mathscinet
    13. A. I. Aptekarev, A. I. Bogolyubskii, M. Yattselev, “Convergence of ray sequences of Frobenius-Padé approximants”, Sb. Math., 208:3 (2017), 313–334  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    14. Bosuwan N., “On the Boundedness of Poles of Generalized Pade Approximants”, Adv. Differ. Equ., 2019, 137  crossref  mathscinet  zmath  isi  scopus
    15. Bosuwan N., “On Row Sequences of Hermite-Pade Approximation and Its Generalizations”, Mathematics, 8:3 (2020)  crossref  isi
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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