Interpolation in a classical Hilbert space of entire functions

Young, Robert M.

doi:10.1090/S0002-9947-1974-0357823-6

Interpolation in a classical Hilbert space of entire functions
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by Robert M. Young PDF

Trans. Amer. Math. Soc. 192 (1974), 97-114 Request permission

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 $|\operatorname {Re} ({\lambda _n}) - n| \leq L \leq (\log 2)/\pi \;( - \infty < n < \infty )$ 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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