  We study the dynamics of a chain of coupled particles subjected to a
restoring force (Klein-Gordon lattice) in the cases of either periodic or
Dirichlet boundary conditions. Precisely, we prove that, when the initial data
are of small amplitude and have long wavelength, the main part of the solution
is interpolated by a solution of the nonlinear Schr\"odinger equation, which in
turn has the property that its Fourier coefficients decay exponentially. The
first order correction to the solution has Fourier coefficients that decay
exponentially in the periodic case, but only as a power in the Dirichlet case.
In particular our result allows one to explain the numerical computations of
the paper \cite{BMP07}.
