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 spacesEp(1 ≤ p ≤ ∞) consisting of all entire functions of exponential type at most π that belong to Lp (−∞, ∞) on the real axis. By virtue of the Paley-Wiener theorem, the Fourier transform establishes an isometric isomorphism between L2(−π, π) andE2. When p is finite, a sequence {λn} of complex numbers will be called aframe forEp 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 inEp. We say that {λn} is aninterpolating sequence forEp if the set of all scalar sequences {f (λn)}, with f eEp, coincides with lp. If in addition {λn} is a set of uniqueness forEp, that is, if the relations f(λn)=0(−∞<n<∞), with f eEp, imply that f ≡0, then we call {λn} acomplete interpolating sequence.