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Interpolation in a Classical Hilbert Space of Entire Functions

Robert M. Young
Transactions of the American Mathematical Society
Vol. 192 (May, 1974), pp. 97-114
DOI: 10.2307/1996822
Stable URL: http://www.jstor.org/stable/1996822
Page Count: 18
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Interpolation in a Classical Hilbert Space of Entire Functions
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Abstract

Let H denote the Paley-Wiener space of entire functions of exponential type π which belong to L2(-∞, ∞) on the real axis. A sequence {λn} 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(λn)}. If in addition {λn} is a set of uniqueness for H, then {λn} 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 λn is sufficiently small, then {λn} is an interpolating sequence. If $|\operatorname{Re} (\lambda_n)- n| \leq L \leq (\log 2)/\pi (-\infty < n < \infty)$ and if the imaginary part of λn is uniformly bounded, then {λn} is a complete interpolating sequence and {eiλnt } is a basis for L2(-π,π). These results are used to investigate interpolating sequences in several related spaces of entire functions of exponential type.

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