Geodesic
Nikishin systems on star-like sets: algebraic properties and weak asymptotics of the associated multiple orthogonal polynomials
Sbornik. Mathematics, Tome 209 (2018) no. 7, pp. 1051-1088 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We investigate polynomials $Q_n(z)$, $n=0,1,\dots$, that are multi-orthogonal with respect to a Nikishin system of $p\geqslant 1 $ compactly supported measures over the star-like set of $p+1$ rays $S_+:=\{z\in \mathbb{C}\colon z^{p+1}\geqslant 0 \}$. We prove that the Nikishin system is normal, that the polynomials satisfy a three-term recurrence relation of order $p+1$ of the form $z Q_{n}(z)=Q_{n+1}(z)+a_{n}Q_{n-p}(z)$ with $a_n>0$ for all $n\geqslant p$, and that the nonzero roots of $Q_n$ are all simple and located in $S_+$. Under the assumption that the measures generating the Nikishin system are regular (in the sense of Stahl and Totik), we describe the asymptotic zero distribution and weak behaviour of the polynomials $Q_n$ in terms of a vector equilibrium problem for logarithmic potentials. Under the same regularity assumptions, we prove a theorem on the convergence of the Hermite-Padé approximants to the Nikishin system of Cauchy transforms. Bibliography: 16 titles.
Keywords: Nikishin system, vector equilibrium problem
Mots-clés : multiple orthogonal polynomials, Hermite-Padé approximation. 
@article{SM_2018_209_7_a5,
     author = {A. L\'opez-Garc{\'\i}a and E. Mi\~na-D{\'\i}az},
     title = {Nikishin systems on star-like sets: algebraic properties and weak asymptotics of the associated multiple orthogonal polynomials},
     journal = {Sbornik. Mathematics},
     pages = {1051--1088},
     year = {2018},
     volume = {209},
     number = {7},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2018_209_7_a5/}
}
TY  - JOUR
AU  - A. López-García
AU  - E. Miña-Díaz
TI  - Nikishin systems on star-like sets: algebraic properties and weak asymptotics of the associated multiple orthogonal polynomials
JO  - Sbornik. Mathematics
PY  - 2018
SP  - 1051
EP  - 1088
VL  - 209
IS  - 7
UR  - http://geodesic.mathdoc.fr/item/SM_2018_209_7_a5/
LA  - en
ID  - SM_2018_209_7_a5
ER  - 
%0 Journal Article
%A A. López-García
%A E. Miña-Díaz
%T Nikishin systems on star-like sets: algebraic properties and weak asymptotics of the associated multiple orthogonal polynomials
%J Sbornik. Mathematics
%D 2018
%P 1051-1088
%V 209
%N 7
%U http://geodesic.mathdoc.fr/item/SM_2018_209_7_a5/
%G en
%F SM_2018_209_7_a5
A. López-García; E. Miña-Díaz. Nikishin systems on star-like sets: algebraic properties and weak asymptotics of the associated multiple orthogonal polynomials. Sbornik. Mathematics, Tome 209 (2018) no. 7, pp. 1051-1088. http://geodesic.mathdoc.fr/item/SM_2018_209_7_a5/

[1] A. I. Aptekarev, V. A. Kalyagin, E. B. Saff, “Higher-order three-term recurrences and asymptotics of multiple orthogonal polynomials”, Constr. Approx., 30:2 (2009), 175–223 | DOI | MR | Zbl

[2] A. I. Aptekarev, V. A. Kaliaguine, J. Van Iseghem, “The genetic sums' representation for the moments of a system of Stieltjes functions and its applications”, Constr. Approx., 16:4 (2000), 487–524 | DOI | MR | Zbl

[3] S. Delvaux, A. López, “High-order three-term recursions, Riemann–Hilbert minors and Nikishin systems on star-like sets”, Constr. Approx., 37:3 (2013), 383–453 | DOI | MR | Zbl

[4] U. Fidalgo Prieto, A. López García, G. López Lagomasino, V. N. Sorokin, “Mixed type multiple orthogonal polynomials for two Nikishin systems”, Constr. Approx., 32:2 (2010), 255–306 | DOI | MR | Zbl

[5] U. Fidalgo Prieto, G. López Lagomasino, “Nikishin systems are perfect”, Constr. Approx., 34:3 (2011), 297–356 | DOI | MR | Zbl

[6] A. A. Gonchar, E. A. Rakhmanov, “Equilibrium measure and the distribution of zeros of extremal polynomials”, Math. USSR-Sb., 53:1 (1986), 119–130 | DOI | MR | Zbl

[7] A. A. Gonchar, E. A. Rakhmanov, V. N. Sorokin, “Hermite–Padé approximants for systems of Markov-type functions”, Sb. Math., 188:5 (1997), 671–696 | DOI | DOI | MR | Zbl

[8] M. X. He, E. B. Saff, “The zeros of Faber polynomials for an $m$-cusped hypocycloid”, J. Approx. Theory, 78:3 (1994), 410–432 | DOI | MR | Zbl

[9] A. López García, “Asymptotics of multiple orthogonal polynomials for a system of two measures supported on a starlike set”, J. Approx. Theory, 163:9 (2011), 1146–1184 | DOI | MR | Zbl

[10] A. López-García, G. López Lagomasino, “Nikishin systems on star-like sets: ratio asymptotics of the associated multiple orthogonal polynomials”, J. Approx. Theory, 225 (2018), 1–40 ; arXiv: 1612.01149 | DOI | MR | Zbl

[11] E. M. Nikišin, “On simultaneous Padé approximants”, Math. USSR-Sb., 41:4 (1982), 409–425 | DOI | MR | Zbl

[12] E. M. Nikishin, V. N. Sorokin, Rational approximations and orthogonality, Transl. Math. Monogr., 92, Amer. Math. Soc., Providence, RI, 1991, viii+221 pp. | DOI | MR | MR | Zbl | Zbl

[13] T. Ransford, Potential theory in the complex plane, London Math. Soc. Stud. Texts, 28, Cambridge Univ. Press, Cambridge, 1995, x+232 pp. | DOI | MR | Zbl

[14] N. Ben Romdhane, “On the zeros of $d$-symmetric $d$-orthogonal polynomials”, J. Math. Anal. Appl., 344:2 (2008), 888–897 | DOI | MR | Zbl

[15] E. B. Saff, V. Totik, Logarithmic potentials with external fields, Grundlehren Math. Wiss., 316, Springer-Verlag, Berlin, 1997, xvi+505 pp. | DOI | MR | Zbl

[16] H. Stahl, V. Totik, General orthogonal polynomials, Encyclopedia Math. Appl., 43, Cambridge Univ. Press, Cambridge, 1992, xii+250 pp. | DOI | MR | Zbl