In this paper the noncharacteristic Cauchy problem (NCP)[formula]is considered. With ϕ∈Lp(R),p∈[1,∞], it is proved that a solution of NCP exists if and only if ϕ is infinitely differentiable and ‖ϕ(n)‖Lp(R)≤c(2n)!s2n, ∀n∈N, for certain constantscands. NCP is well known to be severely ill-posed: a small perturbation in the Cauchy data may cause a dramatically large error in the solution. The following mollification method is suggested for solving NCP in a stable way: If ϕ∈Lp(R) is given inexactly by ϕϵ∈Lp(R) then we mollify ϕϵby convolutions with the Dirichlet kernel and the de la Vallée Poussin kernel. The exact solution of NCP is approximated by the solution of the mollified problem with a reasonable choice of mollification parameters which yields error estimates of the Hölder type. By the method we can work with the data inLp(R),p∈[1,∞] and obtain several sharp stability estimates inLp- andL∞-norms of the Hölder type for the solution of the problem. The method can easily be implemented numerically using the fast Fourier transform. A stable marching difference scheme based on this method is suggested. Several numerical examples are given, which show that the method is effective.