  We consider a nonlinear Klein-Gordon equation with a quasilinear quadratic
term. The Nonlinear Schr\"odinger (NLS) equation can be derived as a formal
approximation equation describing the evolution of the envelopes of slowly
modulated spatially and temporarily oscillating wave packet-like solutions to
the quasilinear Klein-Gordon equation. It is the purpose of this paper to
present a method which allows one to prove error estimates in Sobolev norms
between exact solutions of the quasilinear Klein-Gordon equation and the formal
approximation obtained via the NLS equation. The paper contains the first
validity proof of the NLS approximation of a nonlinear hyperbolic equation with
a quasilinear quadratic term by error estimates in Sobolev spaces. We expect
that the method developed in the present paper will allow an answer to the
relevant question of the validity of the NLS approximation for other
quasilinear hyperbolic systems.
