Interpolation and frames in certain Banach spaces of entire functions

Abstract

The important class of generalized bases known as frames was first introduced by Duffin and Schaeffer in their study of nonharmonic Fourier series in L²(−π, π) [4]. Here we consider more generally the classical Banach spacesE ^p(1 ≤ p ≤ ∞) consisting of all entire functions of exponential type at most π that belong to L^p (−∞, ∞) on the real axis. By virtue of the Paley-Wiener theorem, the Fourier transform establishes an isometric isomorphism between L²(−π, π) andE ². When p is finite, a sequence {λ _n} of complex numbers will be called aframe forE ^p provided the inequalities

$$A\left\| f \right\|^p \leqslant \sum {\left| {f\left( {\lambda _\pi } \right)} \right|^p } \leqslant B\left\| f \right\|^p $$

hold for some positive constants A and B and all functions f inE ^p. We say that {λ _n} is aninterpolating sequence forE ^p if the set of all scalar sequences {f (λ _n)}, with f εE ^p, coincides with ℓ^p. If in addition {λ _n} is a set of uniqueness forE ^p, that is, if the relations f(λ _n)=0(−∞<n<∞), with f εE ^p, imply that f ≡0, then we call {λ _n} acomplete interpolating sequence.

Plancherel and Pólya [7] showed that the integers form a complete interpolating sequence forE ^p whenever1<p<∞. In Section 2 we show that every complete interpolating sequence forE ^p(1<p<∞) remains stable under a very general set of displacements of its elements. In Section 3 we use this result to prove a far-reaching generalization of another classical interpolation theorem due to Ingham [6].