Sampling sets and sufficient sets for \(A^{-\infty}\). (English) Zbl 1019.30026
The Banach space \(A^{-p}\), \(p>0\), of all the holomorphic functions \(f\) on the unit disc \(D\) such that \(\|f\|_p: =\sup_{z\in D} |f(z)|(1-|z|^2)^p <\infty\). The space \(A^{-\infty}\) is the union of \(A^{-p}\), \(p>0\). The authors’ abstract: We give new characterizations of the subsets \(S\) of the unit disc \({\mathbf D}\) of the complex plane such that the topology of the space \(A^{-\infty}\) of holomorphic functions of polynomial growth on \({\mathbf D}\) coincides with the topology of space of the restrictions of the functions to the set \(S\). These sets are called weakly sufficient sets for \(A^{-\infty}\). Our approach is based on a study of the so-called \((p,q)\)-sampling sets which generalize the \(A^{-p}\)-sampling sets of Seip. A characterization of \((p,q)\)-sampling and weakly sufficient rotation invariant sets is included. It permits us to obtain new examples and to solve an open question of Khôi and Thomas.
Reviewer: T.Nakazi (Sapporo)
MSC:
| 30D05 | Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable |
Keywords:
holomorphic functions on the unit disc; sampling sets; polynomial growth; weakly sufficient sets; Bergman spacesReferences:
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