Maximally convergent rational approximants of meromorphic functions

Hans-Peter Blatt

Banach Center Publications, Tome 104 (2015), p. 63-78 / Harvested from The Polish Digital Mathematics Library

Résumé

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 $E_{ρ(f)}$.

Détail

Publié le : 2015-01-01

Zbl 1336.41006

EUDML-ID : urn:eudml:doc:282220