Sbornik: Mathematics
RUS  ENG    JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB  
General information
Latest issue
Forthcoming papers
Archive
Impact factor
Guidelines for authors
License agreement
Submit a manuscript

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Mat. Sb.:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


Sbornik: Mathematics, 2013, Volume 204, Issue 1, Pages 43–74
DOI: https://doi.org/10.1070/SM2013v204n01ABEH004291
(Mi sm8076)
 

This article is cited in 11 scientific papers (total in 11 papers)

A family of Nikishin systems with periodic recurrence coefficients

S. Delvauxa, A. Lópeza, G. López Lagomasinob

a Department of Mathematics, KU Leuven, Belgium
b Departamento de Matemáticas, Universidad Carlos III de Madrid, Spain
References:
Abstract: Suppose we have a Nikishin system of $p$ measures with the $k$th generating measure of the Nikishin system supported on an interval $\Delta_k\subset\mathbb R$ with $\Delta_k\cap\Delta_{k+1}=\varnothing$ for all $k$. It is well known that the corresponding staircase sequence of multiple orthogonal polynomials satisfies a $(p+2)$-term recurrence relation whose recurrence coefficients, under appropriate assumptions on the generating measures, have periodic limits of period $p$. (The limit values depend only on the positions of the intervals $\Delta_k$.) Taking these periodic limit values as the coefficients of a new $(p+2)$-term recurrence relation, we construct a canonical sequence of monic polynomials $\{P_{n}\}_{n=0}^\infty$, the so-called Chebyshev-Nikishin polynomials. We show that the polynomials $P_n$ themselves form a sequence of multiple orthogonal polynomials with respect to some Nikishin system of measures, with the $k$th generating measure being absolutely continuous on $\Delta_k$. In this way we generalize a result of the third author and Rocha [22] for the case $p=2$. The proof uses the connection with block Toeplitz matrices, and with a certain Riemann surface of genus zero. We also obtain strong asymptotics and an exact Widom-type formula for functions of the second kind of the Nikishin system for $\{P_{n}\}_{n=0}^\infty$.
Bibliography: 27 titles.
Keywords: multiple orthogonal polynomial, Nikishin system, block Toeplitz matrix, Hermite-Padé approximant, strong asymptotics, ratio asymptotics.
Funding agency Grant number
Fonds Wetenschappelijk Onderzoek
Ministerio de Ciencia e Innovación de España MTM 2009-12740-C03-01
Received: 16.10.2011 and 13.07.2012
Bibliographic databases:
Document Type: Article
UDC: 517.53
MSC: Primary 42C05; Secondary 41A21
Language: English
Original paper language: Russian
Citation: S. Delvaux, A. López, G. López Lagomasino, “A family of Nikishin systems with periodic recurrence coefficients”, Sb. Math., 204:1 (2013), 43–74
Citation in format AMSBIB
\Bibitem{DelLopLop13}
\by S.~Delvaux, A.~L\'opez, G.~L\'opez Lagomasino
\paper A~family of Nikishin systems with periodic recurrence coefficients
\jour Sb. Math.
\yr 2013
\vol 204
\issue 1
\pages 43--74
\mathnet{http://mi.mathnet.ru//eng/sm8076}
\crossref{https://doi.org/10.1070/SM2013v204n01ABEH004291}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=3060076}
\zmath{https://zbmath.org/?q=an:06197055}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000317573800002}
\elib{https://elibrary.ru/item.asp?id=19066596}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84876730667}
Linking options:
  • https://www.mathnet.ru/eng/sm8076
  • https://doi.org/10.1070/SM2013v204n01ABEH004291
  • https://www.mathnet.ru/eng/sm/v204/i1/p47
  • This publication is cited in the following 11 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Математический сборник Sbornik: Mathematics
    Statistics & downloads:
    Abstract page:640
    Russian version PDF:209
    English version PDF:11
    References:87
    First page:16
     
      Contact us:
     Terms of Use  Registration to the website  Logotypes © Steklov Mathematical Institute RAS, 2024