Abstract
Completing a series of works begun by Wiener [34], Paley and Wiener [28] and Ingham [9], a far-reaching generalization of Parseval"s identity was obtained by Beurling [4] for nonharmonic Fourier series whose exponents satisfy a uniform gap condition. Later this gap condition was weakened by Ullrich [33], Castro and Zuazua [5], Jaffard, Tucsnak and Zuazua [11] and then in [2] 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 [15] and [2]. Finally, applying these results we obtain an optimal simultaneous observability theorem concerning a system of vibrating strings.
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Baiocchi, C., Komornik, V. & Loreti, P. Ingham-Beurling type theorems with weakened gap conditions. Acta Mathematica Hungarica 97, 55–95 (2002). https://doi.org/10.1023/A:1020806811956
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DOI: https://doi.org/10.1023/A:1020806811956