Geodesic
Hermite-Padé approximants to the Weyl function and its derivative for discrete measures
Sbornik. Mathematics, Tome 211 (2020) no. 10, pp. 1486-1502 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Hermite-Padé approximants of the second kind to the Weyl function and its derivatives are investigated. The Weyl function is constructed from the orthogonal Meixner polynomials. The limiting distribution of the zeros of the common denominators of these approximants, which are multiple orthogonal polynomials for a discrete measure, is found. It is proved that the limit measure is the unique solution of the equilibrium problem in the theory of the logarithmic potential with an Angelesco matrix. The effect of pushing some zeros off the real axis to some curve in the complex plane is discovered. An explicit form of the limit measure in terms of algebraic functions is given. Bibliography: 10 titles.
Keywords: Meixner polynomials, equilibrium problems in logarithmic potential theory, Riemann surfaces, algebraic functions. 
@article{SM_2020_211_10_a5,
     author = {V. N. Sorokin},
     title = {Hermite-Pad\'e approximants to the {Weyl} function and its derivative for discrete measures},
     journal = {Sbornik. Mathematics},
     pages = {1486--1502},
     year = {2020},
     volume = {211},
     number = {10},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2020_211_10_a5/}
}
TY  - JOUR
AU  - V. N. Sorokin
TI  - Hermite-Padé approximants to the Weyl function and its derivative for discrete measures
JO  - Sbornik. Mathematics
PY  - 2020
SP  - 1486
EP  - 1502
VL  - 211
IS  - 10
UR  - http://geodesic.mathdoc.fr/item/SM_2020_211_10_a5/
LA  - en
ID  - SM_2020_211_10_a5
ER  - 
%0 Journal Article
%A V. N. Sorokin
%T Hermite-Padé approximants to the Weyl function and its derivative for discrete measures
%J Sbornik. Mathematics
%D 2020
%P 1486-1502
%V 211
%N 10
%U http://geodesic.mathdoc.fr/item/SM_2020_211_10_a5/
%G en
%F SM_2020_211_10_a5
V. N. Sorokin. Hermite-Padé approximants to the Weyl function and its derivative for discrete measures. Sbornik. Mathematics, Tome 211 (2020) no. 10, pp. 1486-1502. http://geodesic.mathdoc.fr/item/SM_2020_211_10_a5/

[1] A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Higher transcendental functions, Based, in part, on notes left by H. Bateman, v. 2, McGraw-Hill Book Company, Inc., New York–Toronto–London, 1953, xvii+396 pp. | MR | MR | Zbl | Zbl

[2] 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

[3] N. S. Landkof, Foundations of modern potential theory, Grundlehren Math. Wiss., 180, Springer-Verlag, New York–Heidelberg, 1972, x+424 pp. | MR | MR | Zbl | Zbl

[4] V. N. Sorokin, “On multiple orthogonal polynomials for discrete Meixner measures”, Sb. Math., 201:10 (2010), 1539–1561 | DOI | DOI | MR | Zbl

[5] A. A. Kandayan, V. N. Sorokin, “Multipoint Hermite–Padé approximations for beta functions”, Math. Notes, 87:2 (2010), 204–217 | DOI | DOI | MR | Zbl

[6] V. N. Sorokin, “Generalized Pollaczek polynomials”, Sb. Math., 200:4 (2009), 577–595 | DOI | DOI | MR | Zbl

[7] V. N. Sorokin, E. N. Cherednikova, “Mnogochleny Meiksnera s peremennym vesom”, Sovremennye problemy matematiki i mekhaniki, VI:1 (2011), 118–125

[8] A. A. Gonchar, E. A. Rakhmanov, “On the equilibrium problem for vector potentials”, Russian Math. Surveys, 40:4 (1985), 183–184 | DOI | MR | Zbl

[9] A. A. Gonchar, E. A. Rakhmanov, S. P. Suetin, “Padé–Chebyshev approximants of multivalued analytic functions, variation of equilibrium energy, and the $S$-property of stationary compact sets”, Russian Math. Surveys, 66:6 (2011), 1015–1048 | DOI | DOI | MR | Zbl

[10] A. Martínez-Finkelshtein, E. A. Rakhmanov, S. P. Suetin, “Variation of the equilibrium energy and the $S$-property of stationary compact sets”, Sb. Math., 202:12 (2011), 1831–1852 | DOI | DOI | MR | Zbl