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Trigonometric integrators for oscillatory linear Hamiltonian differentialequations are considered. Under a condition of Hairer & Lubich on the filterfunctions in the method, a modified energy is derived that is exactly preservedby trigonometric integrators. This implies and extends a known result onall-time near-conservation of energy. The extension can be applied to linearwave equations.

On energy conservation by trigonometric integrators in the linear case with application to wave equations