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Mat. Sb., 2017, Volume 208, Number 3, Pages 4–27 (Mi msb8632)  

This article is cited in 10 scientific papers (total in 10 papers)

Convergence of ray sequences of Frobenius-Padé approximants

A. I. Aptekareva*, A. I. Bogolyubskiib, M. Yattselevc

a Federal Research Centre Keldysh Institute of Applied Mathematics of Russian Academy of Sciences, Moscow
b Russian National Research Medical University named after N. I. Pirogov, Moscow
c Department of Mathematical Sciences, Indiana University – Purdue University Indianapolis, Indianapolis, IN, USA

Abstract: Let $\widehat\sigma$ be a Cauchy transform of a possibly complex-valued Borel measure $\sigma$ and $\{p_n\}$ a system of orthonormal polynomials with respect to a measure $\mu$, where $\operatorname{supp}(\mu)\cap\operatorname{supp}(\sigma)=\varnothing$. An $(m,n)$th Frobenius-Padé approximant to $\widehat\sigma$ is a rational function $P/Q$, $\deg(P)\leq m$, $\deg(Q)\leq n$, such that the first $m+n+1$ Fourier coefficients of the remainder function $Q\widehat\sigma-P$ vanish when the form is developed into a series with respect to the polynomials $p_n$. We investigate the convergence of the Frobenius-Padé approximants to $\widehat\sigma$ along ray sequences $n/(n+m+1)\to c>0$, $n-1\leq m$, when $\mu$ and $\sigma$ are supported on intervals of the real line and their Radon-Nikodym derivatives with respect to the arcsine distribution of the corresponding interval are holomorphic functions.
Bibliography: 30 titles.

Keywords: Frobenius-Padé approximants, linear Padé-Chebyshev approximants, Padé approximants of orthogonal expansions, orthogonality, Markov-type functions, Riemann-Hilbert matrix problem.

Funding Agency Grant Number
Russian Science Foundation 14-21-00025
Russian Foundation for Basic Research 14-01-00604-a
17-01-00614-a
Simons Foundation #354538
A. I. Aptekarev's research was supported by the Russian Science Foundation (grant no. 14-21-00025). A. I. Bogolyubskii's research was supported by the Russian Foundation for Basic Research (grant nos. 14-01-00604_а and 17-01-00614_a). M. L. Yattselev's research was supported by the Simons Foundation (grant #354538).

* Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/sm8632

Full text: PDF file (1155 kB)
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English version:
Sbornik: Mathematics, 2017, 208:3, 313–334

Bibliographic databases:

UDC: 517.53
MSC: 41A20, 41A21
Received: 09.11.2015 and 26.09.2016

Citation: A. I. Aptekarev, A. I. Bogolyubskii, M. Yattselev, “Convergence of ray sequences of Frobenius-Padé approximants”, Mat. Sb., 208:3 (2017), 4–27; Sb. Math., 208:3 (2017), 313–334

Citation in format AMSBIB
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    1. 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
    2. 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
    3. G. López Lagomasino, W. Van Assche, “Riemann-Hilbert analysis for a Nikishin system”, Sb. Math., 209:7 (2018), 1019–1050  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    4. 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  mathscinet  isi  elib  elib
    5. A. P. Starovoitov, “Hermite–Padé approximants of the Mittag-Leffler functions”, Proc. Steklov Inst. Math., 301 (2018), 228–244  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    6. S. P. Suetin, “On an Example of the Nikishin System”, Math. Notes, 104:6 (2018), 905–914  mathnet  crossref  crossref  mathscinet  isi  elib
    7. S. P. Suetin, “Existence of a three-sheeted Nutall surface for a certain class of infinite-valued analytic functions”, Russian Math. Surveys, 74:2 (2019), 363–365  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    8. S. P. Suetin, “Equivalence of a Scalar and a Vector Equilibrium Problem for a Pair of Functions Forming a Nikishin System”, Math. Notes, 106:6 (2019), 970–979  mathnet  crossref  crossref  mathscinet  isi  elib
    9. N. R. Ikonomov, S. P. Suetin, “Scalar Equilibrium Problem and the Limit Distribution of Zeros of Hermite–Padé Polynomials of Type II”, Proc. Steklov Inst. Math., 309 (2020), 159–182  mathnet  crossref  crossref  mathscinet  isi  elib
    10. I A. Bogolyubskii , V. G. Lysov, “Constructive solution of one vector equilibrium problem”, Dokl. Math., 101:2 (2020), 90–92  crossref  isi
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