  The neutrino process that occurs in the outer stellar shells during a
supernova explosion and involves neutrino-nucleus reactions produces a range of
rare, stable and radioactive isotopes. We improve previous $\nu$-process
studies by using, for the first time, the time-dependent neutrino emission
spectra, as predicted from supernova simulations, rather than a simplified
parametric description modeled after the neutron-star cooling phase. In
particular, our calculations use time-dependent neutrino spectra for all
neutrino species, consider their deviation from a Fermi-Dirac distribution and
account for the neutrino emission from the neutrino burst and accretion phases.
We find that the time-dependent treatment of the neutrino emission spectra
results in higher yields for the selected nuclei produced by the $\nu$~process
as compared to previous studies and also compared to the approximation of
assuming constant neutrino energies corresponding to the time-averaged mean
energy radiated in each species. The effect is largest for nuclides produced by
charged-current reactions. Our results reflect the dynamical competition
between neutrino-induced reactions and the effect of the shock passage through
the star. By varying the neutrino burst luminosity and the duration of the
accretion phase, we study the impact of these early emission phases and their
uncertainties on the $\nu$-process nucleosynthesis. We find that the deviation
of the neutrino spectra from a Fermi-Dirac distribution calculated in supernova
simulations has a negligible effect on the $\nu$-process yields.
