Maximally convergent rational approximants of meromorphic functions

• Autorzy: Hans-Peter Blatt
• Języki publikacji: EN

Abstrakty

EN

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 $E_{ρ(f)}$.

Kategorie tematyczne

• 30E10: Approximation in the complex domain
• 41A20: Approximation by rational functions
• 41A25: Rate of convergence, degree of approximation

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Banach Center Publications
• Rocznik: 2015
• Tom: 107
• Numer: 1
• Strony: 63-78

Daty

wydano

2015

Twórcy

autor

Hans-Peter Blatt

• Mathematisch-Geographische Fakultaet, Katholische Universitaet Eichstaett-Ingolstadt, 85071 Eichstaett, Germany

Identyfikatory

DOI

10.4064/bc107-0-5