  For nonlinear wave equations with a potential term we prove pointwise
space-time decay estimates and develop a perturbation theory for small initial
data. We show that the perturbation series has a positive convergence radius by
a method which reduces the wave equation to an algebraic one. We demonstrate
that already first and second perturbation orders, satisfying linear equations,
can provide precise information about the decay of the full solution to the
nonlinear wave equation. In a forthcoming publication (part II) we address the
issue of optimal decay estimates and precise asymptotics under spherical
symmetry where the perturbation equations can be solved almost exactly.
