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 Tr. Mat. Inst. Steklova, 2014, Volume 284, Pages 176–199 (Mi tm3528)

Distribution of zeros of the Hermite–Padé polynomials for a system of three functions, and the Nuttall condenser

R. K. Kovachevaa, S. P. Suetinb

a Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria
b Steklov Mathematical Institute of the Russian Academy of Sciences, Moscow, Russia

Abstract: The well-known approach of J. Nuttall to the derivation of strong asymptotic formulas for the Hermite–Padé polynomials for a set of $m$ multivalued functions is based on the conjecture that there exists a canonical (in the sense of decomposition into sheets) $m$-sheeted Riemann surface possessing certain properties. In this paper, for $m=3$, we introduce a notion of an abstract Nuttall condenser and describe a procedure for constructing (based on this condenser) a three-sheeted Riemann surface $\mathfrak R_3$ that has a canonical decomposition. We consider a system of three functions $\mathfrak f_1,\mathfrak f_2,\mathfrak f_3$ that are rational on the constructed Riemann surface and satisfy the independence condition $\det[\mathfrak f_k(z^{(j)})]\not\equiv0$. In the case of $m=3$, we refine the main theorem from Nuttall's paper of 1981. In particular, we show that in this case the complement $\overline{\mathbb C}\setminus B$ of the open (possibly, disconnected) set $B\subset\overline{\mathbb C}$ introduced in Nuttall's paper consists of a finite number of analytic arcs. We also propose a new conjecture concerning strong asymptotic formulas for the Padé approximants.

 Funding Agency Grant Number Russian Foundation for Basic Research 11-01-00330-a13-01-12430-ofi-m Ministry of Education and Science of the Russian Federation NSh-4664.2012.1 The work of the second author was supported by the Russian Foundation for Basic Research (project nos. 11-01-00330-a and 13-01-12430-ofi-m2) and by a grant of the President of the Russian Federation (project no. NSh-4664.2012.1).

DOI: https://doi.org/10.1134/S0371968514010129

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English version:
Proceedings of the Steklov Institute of Mathematics, 2014, 284, 168–191

Bibliographic databases:

UDC: 517.53

Citation: R. K. Kovacheva, S. P. Suetin, “Distribution of zeros of the Hermite–Padé polynomials for a system of three functions, and the Nuttall condenser”, Function spaces and related problems of analysis, Collected papers. Dedicated to Oleg Vladimirovich Besov, corresponding member of the Russian Academy of Sciences, on the occasion of his 80th birthday, Tr. Mat. Inst. Steklova, 284, MAIK Nauka/Interperiodica, Moscow, 2014, 176–199; Proc. Steklov Inst. Math., 284 (2014), 168–191

Citation in format AMSBIB
\Bibitem{KovSue14} \by R.~K.~Kovacheva, S.~P.~Suetin \paper Distribution of zeros of the Hermite--Pad\'e polynomials for a~system of three functions, and the Nuttall condenser \inbook Function spaces and related problems of analysis \bookinfo Collected papers. Dedicated to Oleg Vladimirovich Besov, corresponding member of the Russian Academy of Sciences, on the occasion of his 80th birthday \serial Tr. Mat. Inst. Steklova \yr 2014 \vol 284 \pages 176--199 \publ MAIK Nauka/Interperiodica \publaddr Moscow \mathnet{http://mi.mathnet.ru/tm3528} \crossref{https://doi.org/10.1134/S0371968514010129} \elib{https://elibrary.ru/item.asp?id=21249111} \transl \jour Proc. Steklov Inst. Math. \yr 2014 \vol 284 \pages 168--191 \crossref{https://doi.org/10.1134/S008154381401012X} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000335559000011} \elib{https://elibrary.ru/item.asp?id=21876711} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84899844590} 

• http://mi.mathnet.ru/eng/tm3528
• https://doi.org/10.1134/S0371968514010129
• http://mi.mathnet.ru/eng/tm/v284/p176

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This publication is cited in the following articles:
1. S. P. Suetin, “Distribution of the zeros of Padé polynomials and analytic continuation”, Russian Math. Surveys, 70:5 (2015), 901–951
2. 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
3. A. Martinez-Finkelshtein, E. .A. Rakhmanov, S. P. Suetin, “Asymptotics of Type I Hermite-Padé Polynomials for Semiclassical Functions”, Modern Trends in Constructive Function Theory, Conference and School on Constructive Functions in honor of Ed Saff's 70th Birthday Location (Vanderbilt Univ, Nashville, TN, 2014), Contemporary Mathematics, 661, 2016, 199–228
4. 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
5. Kovacheva R., “A Note on Overconvergent Subsequences of the Mth Row of Classical Pade Approximants”, AIP Conference Proceedings, 2048, eds. Pasheva V., Popivanov N., Venkov G., Amer Inst Physics, 2018, 050013
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