  Standard numerical integrators suffer from an order reduction when applied to
nonlinear Schr\"{o}dinger equations with low-regularity initial data. For
example, standard Strang splitting requires the boundedness of the solution in
$H^{r+4}$ in order to be second-order convergent in $H^r$, i.e., it requires
the boundedness of four additional derivatives of the solution. We present a
new type of integrator that is based on the variation-of-constants formula and
makes use of certain resonance based approximations in Fourier space. The
latter can be efficiently evaluated by fast Fourier methods. For second-order
convergence, the new integrator requires two additional derivatives of the
solution in one space dimension, and three derivatives in higher space
dimensions. Numerical examples illustrating our convergence results are
included. These examples demonstrate the clear advantage of the Fourier
integrator over standard Strang splitting for initial data with low regularity.
