2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/123650We establish discrete Ingham type and Haraux type inequalities for exponential sums satisfying a weakened gap condition. They enable us to obtain discrete simultaneous observability theorems for systems of vibrating strings or beams.Classical Analysis and ODEsAnalysis of PDEsDiscrete Ingham type inequalities with a weakened gap conditiontext