Maximally convergent rational approximants of meromorphic functions

Hans-Peter Blatt. "Maximally convergent rational approximants of meromorphic functions." Banach Center Publications 107.1 (2015): 63-78. <http://eudml.org/doc/282220>.

@article{Hans2015,
abstract = {Let f be meromorphic on the compact set E ⊂ C with maximal Green domain of meromorphy $E_\{ρ(f)\}$, ρ(f) < ∞. We investigate rational approximants $r_\{n,mₙ\}$ of f on E with numerator degree ≤ n and denominator degree ≤ mₙ. We show that a geometric convergence rate of order $ρ(f)^\{-n\}$ on E implies uniform maximal convergence in m₁-measure inside $E_\{ρ(f)\}$ if mₙ = o(n/log n) as n → ∞. If mₙ = o(n), n → ∞, then maximal convergence in capacity inside $E_\{ρ(f)\}$ can be proved at least for a subsequence Λ ⊂ ℕ. Moreover, an analogue of Walsh’s estimate for the growth of polynomial approximants is proved for $r_\{n,mₙ\}$ outside $E_\{ρ(f)\}$.},
author = {Hans-Peter Blatt},
journal = {Banach Center Publications},
keywords = {rational approximation; convergence in -measure and in capacity},
language = {eng},
number = {1},
pages = {63-78},
title = {Maximally convergent rational approximants of meromorphic functions},
url = {http://eudml.org/doc/282220},
volume = {107},
year = {2015},
}

TY - JOUR
AU - Hans-Peter Blatt
TI - Maximally convergent rational approximants of meromorphic functions
JO - Banach Center Publications
PY - 2015
VL - 107
IS - 1
SP - 63
EP - 78
AB - Let f be meromorphic on the compact set E ⊂ C with maximal Green domain of meromorphy $E_{ρ(f)}$, ρ(f) < ∞. We investigate rational approximants $r_{n,mₙ}$ of f on E with numerator degree ≤ n and denominator degree ≤ mₙ. We show that a geometric convergence rate of order $ρ(f)^{-n}$ on E implies uniform maximal convergence in m₁-measure inside $E_{ρ(f)}$ if mₙ = o(n/log n) as n → ∞. If mₙ = o(n), n → ∞, then maximal convergence in capacity inside $E_{ρ(f)}$ can be proved at least for a subsequence Λ ⊂ ℕ. Moreover, an analogue of Walsh’s estimate for the growth of polynomial approximants is proved for $r_{n,mₙ}$ outside $E_{ρ(f)}$.
LA - eng
KW - rational approximation; convergence in -measure and in capacity
UR - http://eudml.org/doc/282220
ER -