ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

8. Analysis

1991 MS Classification: 30, 32, 41 Radyno, Alexander, Belarusian State University, Belarus The degree of rational approximation of a holomorphic function Let E = {z : |z| ≤ 1}, F = {z : |z| ≥ ρ > 1} and f be a holomorphic function on D_{ρ} = C̄ \ F with finite positive order σ:

σ = lim sup_{r→ρ} \frac{ln^{+} ln^{+} max_{|z|=r} |f(z)|}{− ln(ρ − r)}.

Let ρ_{n,m} be the distance of f in the uniform metric on E from the class of all rational functions of order at most (n, m). The results characterizing the rate of convergence to zero of the sequence {ρ_{n,m(n)}}_{n=0}^{∞}, m(n)/n → θ ∈ (0, 1] as n → ∞ in terms of σ are obtained. Namely, the following upper estimates are valid:

lim sup_{n→∞} \frac{ln^{+} ln^{+} (∏_{j=0}^{m(n)} ρ_{n−j,m(n)−j}ρ^{θn^{2}})}{ln n} ≤ 1 + \frac{σ}{σ + 1},

lim inf_{n→∞} \frac{ln^{+} ln^{+} (ρ_{n,m(n)}ρ^{2n/(2−θ)})}{ln n} ≤ \frac{σ}{σ + 1}.