Interpolation in a classical Hilbert space of entire functions

Abstract

Let H denote the Paley-Wiener space of entire functions of exponential type π \pi which belong to L 2 ( − ∞ , ∞ ) {L^2}( - \infty ,\infty ) on the real axis. A sequence { λ n } \{ {\lambda _n}\} of distinct complex numbers will be called an interpolating sequence for H if T H ⊃ l 2 TH \supset {l^2} , where T is the mapping defined by T f = { f ( λ n ) } Tf = \{ f({\lambda _n})\} . If in addition { λ n } \{ {\lambda _n}\} is a set of uniqueness for H, then { λ n } \{ {\lambda _n}\} is called a complete interpolating sequence. The following results are established. If Re ⁡ ( λ n + 1 ) − Re ⁡ ( λ n ) ≥ γ > 1 \operatorname {Re} ({\lambda _{n + 1}}) - \operatorname {Re} ({\lambda _n}) \geq \gamma > 1 and if the imaginary part of λ n {\lambda _n} is sufficiently small, then { λ n } \{ {\lambda _n}\} is an interpolating sequence. If | Re ⁡ ( λ n ) − n | ≤ L ≤ ( log ⁡ 2 ) / π ( − ∞ > n > ∞ ) |\operatorname {Re} ({\lambda _n}) - n| \leq L \leq (\log 2)/\pi \;( - \infty > n > \infty ) and if the imaginary part of λ n {\lambda _n} is uniformly bounded, then { λ n } \{ {\lambda _n}\} is a complete interpolating sequence and { e i λ n t } \{ {e^{i{\lambda _n}t}}\} is a basis for L 2 ( − π , π ) {L^2}( - \pi ,\pi ) . These results are used to investigate interpolating sequences in several related spaces of entire functions of exponential type.