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

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

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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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• https://doi.org/10.4213/sm8076
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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. 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
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
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
4. S. P. Suetin, “Distribution of the zeros of Padé polynomials and analytic continuation”, Russian Math. Surveys, 70:5 (2015), 901–951
5. A. V. Komlov, S. P. Suetin, “Distribution of the zeros of Hermite–Padé polynomials”, Russian Math. Surveys, 70:6 (2015), 1179–1181
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
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
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
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
10. Lopez-Garcia A. Lopez Lagomasino G., “Nikishin Systems on Star-Like Sets: Ratio Asymptotics of the Associated Multiple Orthogonal Polynomials, II”, J. Approx. Theory, 250 (2020), UNSP 105320
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