# Maximally convergent rational approximants of meromorphic functions

Banach Center Publications (2015)

- Volume: 107, Issue: 1, page 63-78
- ISSN: 0137-6934

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topHans-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 -

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