Direct and inverse theorems of rational approximation in the Bergman space

T. S. Mardvilko ; A. A. Pekarskii

Sbornik. Mathematics, Tome 202 (2011) no. 9, pp. 1327-1346

Résumé

For positive numbers $p$ and $\mu$ let $A_{p,\mu}$ denote the Bergman space of analytic functions in the half-plane $\Pi:=\{z\in\mathbb C:\operatorname{Im} z>0\}$. For $f\in A_{p,\mu}$ let $R_n (f)_{p,\mu}$ be the best approximation by rational functions of degree at most $n$. Also let $\alpha\in\mathbb R$ and $\tau>0$ be numbers such that $\alpha+\mu=\frac{1}{\tau}-\frac{1}{p}>0$ and $\frac{1}{p}+\mu\notin\mathbb N$. Then the main result of the paper claims that the set of functions $f\in A_{p,\mu}$ such that $$ \sum_{n=1}^\infty\frac{1}{n}(n^{\alpha+\mu} R_n (f)_{p,\mu})^\tau\infty $$ is precisely the Besov space $B_\tau^\alpha$ of analytic functions in $\Pi$. Bibliography: 23 titles.

Détail

Keywords: direct and inverse theorems of rational approximation, Bernstein-type inequalities, Jackson-type inequalities, Bergman spaces
Mots-clés : Besov spaces.