  We derive dispersion estimates for solutions of the one-dimensional discrete
perturbed Schr\"odinger and wave equations. In particular, we improve upon
previous works and weaken the conditions on the potentials. To this end we also
provide new results concerning scattering for one-dimensional discrete
perturbed Schr\"odinger operators which are of independent interest. Most
notably we show that the reflection and transmission coefficients belong to the
Wiener algebra.
