Interpolation and frames in certain Banach spaces of entire functions

The important class of generalized bases known as frames was first introduced by Duffin and Schaeffer in their study of nonharmonic Fourier series in L-2(-pi,pi) [4]. Here we consider more generally the classical Banach spaces E-P(1 less than or equal to p less than or equal to infinity) consisting of all entire functions of exponential type at most ir that belong to L-p(-infinity,infinity) on the real axis. By virtue of the Paley-Wiener theorem, the Fourier transform establishes an isometric isomorphism between L-2(-pi,pi) and E-2. When p is finite, a sequence (lambda(n)) of complex numbers will be called a frame for E-p provided the inequalities A\\f\\(p) less than or equal to Sigma\f(lambda(n))\(p) less than or equal to B\\f\\(p) hold for some positive constants A and B and all functions f in E-p. We say that (lambda(n)) is an interpolating sequence for E-p if the set of all scalar sequences {f(lambda(n))}, with f epsilon E-P, coincides with l(p). If in addition (lambda(n)) is a set of uniqueness for E-p, that is, if the relations f(lambda(n)) = 0(-infinity < n < infinity), with f epsilon E-p, imply that f = 0, then we call (lambda(n)) a complete interpolating sequence. Plancherel and Polya [7] showed that the integers form a complete interpolating sequence for E-p whenever 1 < p < infinity. In Section 2 we show that every complete interpolating sequence for E-p(1 < p < infinity) 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].