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Mat. Sb., 2009, Volume 200, Number 7, Pages 39–106 (Mi msb4090)  

This article is cited in 7 scientific papers (total in 7 papers)

An analogue of Fabry's theorem for generalized Padé approximants

V. I. Buslaev

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: The current theory of Padé approximation emphasises results of an inverse character, when conclusions about the properties of the approximated function are drawn from information about the behaviour of the approximants. In this paper Gonchar's conjecture is proved; it states that analogues of Fabry's classical ‘ratio’ theorem hold for rows of the table of Padé approximants for orthogonal expansions, multipoint Padé approximants and Padé-Faber approximants. These are the most natural generalizations of the construction of classical Padé approximants. For these Gonchar's conjecture has already been proved by Suetin. The proof presented here is based, on the one hand, on Suetin's result and, on the other hand, on an extension of Poincaré's theorem on recurrence relations with coefficients constant in the limit, which is obtained in the paper.
Bibliography: 19 titles.

Keywords: Padé approximants, recurrence relations, Fabry's theorem, orthogonal polynomials, Faber polynomials.

DOI: https://doi.org/10.4213/sm4090

Full text: PDF file (1012 kB)
References: PDF file   HTML file

English version:
Sbornik: Mathematics, 2009, 200:7, 981–1050

Bibliographic databases:

UDC: 517.535
MSC: Primary 30E10, 41A27; Secondary 41A21
Received: 15.11.2007 and 18.03.2009

Citation: V. I. Buslaev, “An analogue of Fabry's theorem for generalized Padé approximants”, Mat. Sb., 200:7 (2009), 39–106; Sb. Math., 200:7 (2009), 981–1050

Citation in format AMSBIB
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    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. 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
    2. Bosuwan N., Lopez Lagomasino G., “Inverse Theorem on Row Sequences of Linear Pade-Orthogonal Approximation”, Comput. Methods Funct. Theory, 15:4, SI (2015), 529–554  crossref  mathscinet  zmath  isi  elib  scopus
    3. Bosuwan N., “Direct and Inverse Results on Row Sequences of Generalized Pade Approximants to Polynomial Expansions”, Acta Math. Hung., 157:1 (2019), 191–219  crossref  mathscinet  zmath  isi  scopus
    4. Bosuwan N., Lopez Lagomasino G., “Direct and Inverse Results on Row Sequences of Simultaneous Pade-Faber Approximants”, Mediterr. J. Math., 16:2 (2019), 36  crossref  mathscinet  zmath  isi  scopus
    5. Bosuwan N., “On the Boundedness of Poles of Generalized Pade Approximants”, Adv. Differ. Equ., 2019, 137  crossref  mathscinet  zmath  isi  scopus
    6. Matos J.C., Matos J.A., Rodrigues M.J., Vasconcelos P.B., “Approximating the Solution of Integro-Differential Problems Via the Spectral Tau Method With Filtering”, Appl. Numer. Math., 149:SI (2020), 164–175  crossref  mathscinet  zmath  isi
    7. Bosuwan N., “On Row Sequences of Hermite-Pade Approximation and Its Generalizations”, Mathematics, 8:3 (2020)  crossref  isi
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