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Transactions of the American Mathematical Society

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Interpolation in a classical Hilbert space of entire functions


Author: Robert M. Young
Journal: Trans. Amer. Math. Soc. 192 (1974), 97-114
MSC: Primary 30A98; Secondary 30A80, 46E20
DOI: https://doi.org/10.1090/S0002-9947-1974-0357823-6
MathSciNet review: 0357823
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Abstract: Let H denote the Paley-Wiener space of entire functions of exponential type $ \pi $ which belong to $ {L^2}( - \infty ,\infty )$ on the real axis. A sequence $ \{ {\lambda _n}\} $ of distinct complex numbers will be called an interpolating sequence for H if $ TH \supset {l^2}$, where T is the mapping defined by $ Tf = \{ f({\lambda _n})\} $. If in addition $ \{ {\lambda _n}\} $ is a set of uniqueness for H, then $ \{ {\lambda _n}\} $ is called a complete interpolating sequence. The following results are established. If $ \operatorname{Re} ({\lambda _{n + 1}}) - \operatorname{Re} ({\lambda _n}) \geq \gamma > 1$ and if the imaginary part of $ {\lambda _n}$ is sufficiently small, then $ \{ {\lambda _n}\} $ is an interpolating sequence. If $ \vert\operatorname{Re} ({\lambda _n}) - n\vert \leq L \leq (\log 2)/\pi \;( - \infty < n < \infty )$ and if the imaginary part of $ {\lambda _n}$ is uniformly bounded, then $ \{ {\lambda _n}\} $ is a complete interpolating sequence and $ \{ {e^{i{\lambda _n}t}}\} $ is a basis for $ {L^2}( - \pi ,\pi )$. These results are used to investigate interpolating sequences in several related spaces of entire functions of exponential type.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1974-0357823-6
Keywords: Paley-Wiener space, interpolating sequence, complete interpolating sequence, nonharmonic Fourier series expansion, entire functions of exponential type
Article copyright: © Copyright 1974 American Mathematical Society

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