Geodesic
The polynomial Hermite-Pad\'e $m$-system for meromorphic functions on a~compact Riemann surface
Sbornik. Mathematics, Tome 212 (2021) no. 12, pp. 1694-1729

Voir la notice de l'article provenant de la source Math-Net.Ru

Given a tuple of $m+1$ germs of arbitrary analytic functions at a fixed point, we introduce the polynomial Hermite-Padé $m$-system, which includes the Hermite-Padé polynomials of types I and II. In the generic case we find the weak asymptotics of the polynomials of the Hermite-Padé $m$-system constructed from the tuple of germs of functions $1, f_1,\dots,f_m$ that are meromorphic on an $(m+1)$-sheeted compact Riemann surface $\mathfrak R$. We show that if $f_j = f^j$ for some meromorphic function $f$ on $\mathfrak R$, then with the help of the ratios of polynomials of the Hermite-Padé $m$-system we recover the values of $f$ on all sheets of the Nuttall partition of $\mathfrak R$, apart from the last sheet. Bibliography: 18 titles.
Keywords: rational approximation, Hermite-Padé polynomials, weak asymptotics, Riemann surface. 
@article{SM_2021_212_12_a2,
     author = {A. V. Komlov},
     title = {The polynomial {Hermite-Pad\'e} $m$-system for meromorphic functions on a~compact {Riemann} surface},
     journal = {Sbornik. Mathematics},
     pages = {1694--1729},
     publisher = {mathdoc},
     volume = {212},
     number = {12},
     year = {2021},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2021_212_12_a2/}
}
TY  - JOUR
AU  - A. V. Komlov
TI  - The polynomial Hermite-Pad\'e $m$-system for meromorphic functions on a~compact Riemann surface
JO  - Sbornik. Mathematics
PY  - 2021
SP  - 1694
EP  - 1729
VL  - 212
IS  - 12
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SM_2021_212_12_a2/
LA  - en
ID  - SM_2021_212_12_a2
ER  - 
%0 Journal Article
%A A. V. Komlov
%T The polynomial Hermite-Pad\'e $m$-system for meromorphic functions on a~compact Riemann surface
%J Sbornik. Mathematics
%D 2021
%P 1694-1729
%V 212
%N 12
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SM_2021_212_12_a2/
%G en
%F SM_2021_212_12_a2
A. V. Komlov. The polynomial Hermite-Pad\'e $m$-system for meromorphic functions on a~compact Riemann surface. Sbornik. Mathematics, Tome 212 (2021) no. 12, pp. 1694-1729. http://geodesic.mathdoc.fr/item/SM_2021_212_12_a2/