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On the approximability in certain regions by linear combinations of derivatives of analytic functions. (Über die Approximierbarkeit in gewissen Gebieten durch Linearkombinationen von Ableitungen analytischer Funktionen.) (Russian) Zbl 0148.05303

Let \(B\) denote the space of functions analytic in \(\vert z\vert > r\) and vanishing at \(\infty\), and \(A\) the space of functions analytic in \(r <\vert z\vert < R\), the topology being that of uniform convergence on compact subsets of \(\vert z\vert > r\) or \(r < \vert z\vert < R\), respectively. The author shows that the sequence of derivatives \(\{f^{(n)}(z)\}\), \(n = 0, 1, 2, \ldots,\) of a fixed function \(f(z)\in B\) is complete in \(B\) if and only if \(f(z) = k/(z - \alpha)\), \(\alpha\), \(k\) constant, \(\vert \alpha\vert \le r\), \(k\ne 0\). On the other hand a sequence \(\{f^{(n)}(z)\}\), \(f\in A\), is never complete in \(A\).

MSC:

30E10 Approximation in the complex plane
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