On a theorem of Adamian, Arov, and Krein

V. A. Prokhorov

Sbornik. Mathematics, Tome 78 (1994) no. 1, pp. 77-90

Résumé

Some questions in the theory of Hankel operators are considered. The basic results include a theorem generalizing the Adamian–Arov–Krein theorem for the case when the continuous function $f$ giving rise to the Hankel operator $A_f$ is defined on the boundary of a multiply connected domain $G$ bounded by finitely many closed analytic Jordan curves $\Gamma$. Estimates are obtained for the singular numbers $s_n$ of the Hankel operator $A_f$ in terms of the best approximation $\Delta_n$ of $f$ in the space $L_\infty(\Gamma)$ by functions belonging to the class $\mathcal R_n+E_\infty(G)$, where $\mathcal R_n$ is the class of rational functions of order at most $n$, and $E_\infty(G)$ is the Smirnov class of bounded analytic functions on $G$.