Classes of analytic functions determined by best rational approximations in~$H_p$

A. A. Pekarskii

Sbornik. Mathematics, Tome 55 (1986) no. 1, pp. 1-18

Résumé

Let $R_n(f,H_p)$ be the best approximation to the function $f$ in the Hardy space $H_p$ by rational functions of degree at most $n-1$. It is shown that, for example, $f\in H_p$ ($1$) satisfies the condition $\sum_{k=0}^\infty(2^{k\alpha}R_{2^k}(f,H_p))^\sigma\infty$ ($\alpha>0$, $\sigma=(\alpha+p^{-1})^{-1}$) if and only if $f$ belongs to the Hardy–Besov space $B_\sigma^\alpha$. Rational approximation is also considered in $H_p$ ($p\leqslant1$) and $H_\infty$. Some applications of the results are given. Bibliography: 29 titles.