Generalization of some theorems of Ingham and Beurling and applications to control theory

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