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On a theorem of Adamian, Arov, and Krein

V. A. Prokhorov

Belarusian State University

Abstract: 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 $G$.

UDC: 517.5

MSC: Primary 47B35, 41A25, 41A20; Secondary 30E10, 30H05

Received: 10.10.1991 and 25.06.1992

 English version: Russian Academy of Sciences. Sbornik. Mathematics, 1994, 78:1, 77–90

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