Generalization of some theorems of Ingham and Beurling and applications tocontrol theory

Following some classical works of Paley and Wienes Ingham and Beurling gene ralized Parseval's identity to nonharmonic Fourier series whose exponents a re uniformly discrete. Motivated by control-theoretical problems, the last hypothesis was recently weakened in particular cases by Castro, Jaffard, Tu csnak and Zuazua. Generalizing another method, due to Kahane, we prove a ge neral theorem containing all previous results, its efficiency is illustrate d by applications to control theory. (C) 2000 Academie des sciences/Edition s scientifiques et medicales Elsevier SAS.