Article
Ufa Mathematical Journal
Volume 3, Number 1, pp. 30-41
On an orhosimilar system in the space of analytical function and a problem of describing the dual space.
Napalkov V.V.
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We consider an orthosimilar system with measure $\mu$ in the space of analitical function $H$ on the domain $G\subset {\mathbb C}$. Let $K_H(\xi,t),\,\xi,t\in G$ be a reproduction kernel the space $H$. We claim that a system $\{K_H(\xi,t)\}_{t\in G}$ be the orthosimilar system with measure $\mu$ in the space $H$ if and only if the space $H$ coincides with the space $B_2(G,\mu)$ . A problem of describing of the dual space in terms of Hilbert transform is considered. This problem is equivalent to the problem as to when there exists special orthosimilar system in $B_2(G,\mu)$. We prove that the space $\widetilde B_2(G,\mu)$ is uniquely reproduction kernel's Hilbert space of functions on the domain ${\mathbb C}\backslash \overline$ with orthosimilar system $\{\tfrac{1}{(z-\xi)^2}\}_{\xi\in G}$ with measure $\mu$.