INTERPOLATION AND FRAMES IN CERTAIN BANACH-SPACES OF ENTIRE-FUNCTIONS

The important class of generalized bases known as frames was first int roduced by Duffin and Schaeffer in their study of nonharmonic Fourier series in L-2(-pi,pi) [4]. Here we consider more generally the classic al Banach spaces E-P(1 less than or equal to p less than or equal to i nfinity) consisting of all entire functions of exponential type at mos t ir that belong to L-p(-infinity,infinity) on the real axis. By virtu e of the Paley-Wiener theorem, the Fourier transform establishes an is ometric isomorphism between L-2(-pi,pi) and E-2. When p is finite, a s equence (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(lamb da(n))\(p) less than or equal to B\\f\\(p) hold for some positive cons tants A and B and all functions f in E-p. We say that (lambda(n)) is a n 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 in terpolating sequence for E-p whenever 1 < p < infinity. In Section 2 w e show that every complete interpolating sequence for E-p(1 < p < infi nity) remains stable under a very general set of displacements of its elements. In Section 3 we use this result to prove a far-reaching gene ralization of another classical interpolation theorem due to Ingham [6 ].