Geodesic
High-order recurrence relations, Hermite-Padé approximation and Nikishin systems
Sbornik. Mathematics, Tome 209 (2018) no. 3, pp. 385-420 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The study of sequences of polynomials satisfying high-order recurrence relations is connected with the asymptotic behaviour of multiple orthogonal polynomials, the convergence properties of type II Hermite-Padé approximation and eigenvalue distribution of banded Toeplitz matrices. We present some results for the case of recurrences with constant coefficients which match what is known for the Chebyshev polynomials of the first kind. In particular, under appropriate assumptions, we show that the sequence of polynomials satisfies multiple orthogonality relations with respect to a Nikishin-type system of measures. Bibliography: 20 titles.
Keywords: high-order recurrence relation, multiple orthogonality, Nikishin system.
Mots-clés : Hermite-Padé approximation 
@article{SM_2018_209_3_a4,
     author = {D. Barrios Rolan{\'\i}a and J. S. Geronimo and G. L\'opez Lagomasino},
     title = {High-order recurrence relations, {Hermite-Pad\'e} approximation and {Nikishin} systems},
     journal = {Sbornik. Mathematics},
     pages = {385--420},
     year = {2018},
     volume = {209},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2018_209_3_a4/}
}
TY  - JOUR
AU  - D. Barrios Rolanía
AU  - J. S. Geronimo
AU  - G. López Lagomasino
TI  - High-order recurrence relations, Hermite-Padé approximation and Nikishin systems
JO  - Sbornik. Mathematics
PY  - 2018
SP  - 385
EP  - 420
VL  - 209
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/SM_2018_209_3_a4/
LA  - en
ID  - SM_2018_209_3_a4
ER  - 
%0 Journal Article
%A D. Barrios Rolanía
%A J. S. Geronimo
%A G. López Lagomasino
%T High-order recurrence relations, Hermite-Padé approximation and Nikishin systems
%J Sbornik. Mathematics
%D 2018
%P 385-420
%V 209
%N 3
%U http://geodesic.mathdoc.fr/item/SM_2018_209_3_a4/
%G en
%F SM_2018_209_3_a4
D. Barrios Rolanía; J. S. Geronimo; G. López Lagomasino. High-order recurrence relations, Hermite-Padé approximation and Nikishin systems. Sbornik. Mathematics, Tome 209 (2018) no. 3, pp. 385-420. http://geodesic.mathdoc.fr/item/SM_2018_209_3_a4/

[1] A. I. Aptekarev, V. A. Kalyagin, “Analytic properties of two-dimensional continued P-fraction expansions with periodical coefficients and their simultaneous Padé–Hermite approximants”, Rational approximation and applications in mathematics and physics (Łańcut, 1985), Lecture Notes in Math., 1237, Springer, Berlin, 1987, 145–160 | DOI | MR | Zbl

[2] A. I. Aptekarev, G. López Lagomasino, I. A. Rocha, “Ratio asymptotics of Hermite–Padé polynomials for Nikishin systems”, Sb. Math., 196:8 (2005), 1089–1107 | DOI | DOI | MR | Zbl

[3] 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

[4] A. Böttcher, S. M. Grudsky, Spectral properties of banded Toeplitz matrices, SIAM, Philadelphia, PA, 2005, x+411 pp. | DOI | MR | Zbl

[5] E. Coussement, J. Coussement, W. Van Assche, “Asymptotic zero distribution for a class of multiple orthogonal polynomials”, Trans. Amer. Math. Soc., 360:10 (2008), 5571–5588 | DOI | MR | Zbl

[6] 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

[7] S. Delvaux, A. López, G. López Lagomasino, “A family of Nikishin systems with periodic recurrence coefficients”, Sb. Math., 204:1 (2013), 43–74 | DOI | DOI | MR | Zbl

[8] M. Duits, A. B. J. Kuijlaars, “An equillibrium problem for the limiting eigenvalue distribution of banded Toeplitz matrices”, SIAM J. Matrix Anal. Appl., 30:1 (2008), 173–196 | DOI | MR | Zbl

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

[10] 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

[11] V. A. Kalyagin, “Hermite–Padé approximants and spectral analysis of nonsymmetric operators”, Russian Acad. Sci. Sb. Math., 82:1 (1995), 199–216 | DOI | MR | Zbl

[12] V. Kaliaguine, “The operator moment problem, vector continued fractions and an explicit form of the Favard theorem for vector orthogonal polynomials”, J. Comput. Appl. Math., 65:1-3 (1995), 181–193 | DOI | MR | Zbl

[13] A. B. J. Kuijlaars, P. Román, “Recurrence relations and vector equilibrium problems arising from a model of non-intersecting squared Bessel paths”, J. Approx. Theory, 162:11 (2010), 2048–2077 | DOI | MR | Zbl

[14] 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

[15] G. López Lagomasino, I. A. Rocha, “A canonical family of multiple orthogonal polynomials for Nikishin systems”, J. Math. Anal. Appl., 372:2 (2010), 390–401 | DOI | MR | Zbl

[16] R. Miranda, Algeraic curves and Riemann surfaces, Grad. Stud. Math., 5, Amer. Math. Soc., Providence, RI, 1995, xxii+390 pp. | DOI | MR | Zbl

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

[18] V. I. Parusnikov, “The Jacobi–Perron algorithm and simultaneous approximation of functions”, Math. USSR-Sb., 42:2 (1982), 287–296 | DOI | MR | Zbl

[19] P. Schmidt, F. Spitzer, “The Toeplitz matrices of an arbitrary Laurent polynomial”, Math. Scand., 8 (1960), 15–38 | DOI | MR | Zbl

[20] J. Van Iseghem, “Vector orthogonal relations. Vector QD-algorithm”, J. Comput. Appl. Math., 19:1 (1987), 141–150 | DOI | MR | Zbl