Uniform observability of the wave equation via a discrete Ingham inequality

In this paper we prove a discrete version of the classical Ingham inequality for nonharmonic Fourier series whose exponents satisfy a gap condition. Time integrals are replaced by discrete sums on a discrete mesh. We prove that, as the mesh becomes finer and finer, the limit of the discrete Ingham inequality is the classical continuous one. This analysis is partially motivated by control-theoretical applications. As an application we analyze the observation properties of numerical approximation schemes of the 1-d wave equation. The discrete Ingham inequality provides observability (and controllability) results which are uniform with respect to the mesh size in suitable classes of numerical solutions in which the high frequency components have been filtered. We also discuss the optimality of these results in connection with the dispersion diagrams of the considered numerical schemes.