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Mat. Sb., 2013, Volume 204, Number 1, Pages 47–78 (Mi msb8076)  

This article is cited in 9 scientific papers (total in 9 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

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


DOI: https://doi.org/10.4213/sm8076

Full text: PDF file (723 kB)
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English version:
Sbornik: Mathematics, 2013, 204:1, 43–74

Bibliographic databases:

UDC: 517.53
MSC: Primary 42C05; Secondary 41A21
Received: 16.10.2011 and 13.07.2012

Citation: S. Delvaux, A. López, G. López Lagomasino, “A family of Nikishin systems with periodic recurrence coefficients”, Mat. Sb., 204:1 (2013), 47–78; Sb. Math., 204:1 (2013), 43–74

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Delvaux S., López A., “Abey High-order three-term recursions, Riemann–Hilbert minors and Nikishin systems on star-like sets”, Constr. Approx., 37:3 (2013), 383–453  crossref  mathscinet  zmath  isi  elib  scopus
    2. R. K. Kovacheva, S. P. Suetin, “Distribution of zeros of the Hermite–Padé polynomials for a system of three functions, and the Nuttall condenser”, Proc. Steklov Inst. Math., 284 (2014), 168–191  mathnet  crossref  crossref  isi  elib  elib
    3. V. I. Buslaev, S. P. Suetin, “On equilibrium problems related to the distribution of zeros of the Hermite–Padé polynomials”, Proc. Steklov Inst. Math., 290:1 (2015), 256–263  mathnet  crossref  crossref  isi  elib  elib
    4. S. P. Suetin, “Distribution of the zeros of Padé polynomials and analytic continuation”, Russian Math. Surveys, 70:5 (2015), 901–951  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    5. A. V. Komlov, S. P. Suetin, “Distribution of the zeros of Hermite–Padé polynomials”, Russian Math. Surveys, 70:6 (2015), 1179–1181  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    6. W. Van Assche, “Ratio asymptotics for multiple orthogonal polynomials”, Modern trends in constructive function theory, Contemp. Math., 661, ed. D. Hardin, D. Lubinsky, B. Simanek, Amer. Math. Soc., Providence, RI, 2016, 73–85  crossref  mathscinet  zmath  isi
    7. 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, Contemp. Math., 661, ed. D. Hardin, D. Lubinsky, B. Simanek, Amer. Math. Soc., Providence, RI, 2016, 199–228  crossref  mathscinet  zmath  isi
    8. A. Lopez-Garcia, G. Lopez Lagomasino, “Nikishin systems on star-like sets: ratio asymptotics of the associated multiple orthogonal polynomials”, J. Approx. Theory, 225 (2018), 1–40  crossref  mathscinet  zmath  isi  scopus
    9. D. Barrios Rolanía, J. S. Geronimo, G. López Lagomasino, “High-order recurrence relations, Hermite-Padé approximation and Nikishin systems”, Sb. Math., 209:3 (2018), 385–420  mathnet  crossref  crossref  adsnasa  isi  elib
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