Authors:Claudio Baiocchi, Vilmos Komornik, and Paola Loreti
Completing a series of works begun by Wiener , Paley and Wiener  and Ingham , a far-reaching generalization of
Parseval"s identity was obtained by Beurling  for nonharmonic Fourier series whose exponents satisfy a uniform gap condition.
Later this gap condition was weakened by Ullrich , Castro and Zuazua , Jaffard, Tucsnak and Zuazua  and then in
 in some particular cases. In this paper we prove a general theorem which contains all previous results. Furthermore, applying
a different method, we prove a variant of this theorem for nonharmonic Fourier series with vector coefficients. This result, partly motivated by control-theoretical applications, extends several earlier results obtained
in  and . Finally, applying these results we obtain an optimal simultaneous observability theorem concerning a system
of vibrating strings.