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Density theorems for sampling and interpolation in the Bargmann-Fock space. III. (English) Zbl 0789.30024

Let \(\Gamma=\{z_ j\}^ \infty_{j=1}\) be a sequence of distinct complex numbers and associate with it a “multiplicity function” \(\nu\colon \Gamma \to \mathbb{N}\). A sampling and interpolation problem is formulated for the Bargmann-Fock space of entire functions, involving the \(\nu(z_ j)-1\) first derivatives at each point \(z_ j \in \Gamma\). Assuming \(\displaystyle\sup_{z \in \Gamma} \nu(z)<\infty\), a complete description of such sampling and interpolation is given, based on a density concept of Beurling.
The theorems obtained are natural extensions of the results of parts I and II of this series of papers [see K. Seip, J. Reine Angew. Math. 429, 91–106 (1992; Zbl 0745.46034); K. Seip and R. Wallstén, J. Reine Angew. Math. 429, 107–113 (1992; Zbl 0745.46033)].

MSC:

46E20 Hilbert spaces of continuous, differentiable or analytic functions
30E05 Moment problems and interpolation problems in the complex plane
30H05 Spaces of bounded analytic functions of one complex variable
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
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