  We determine the evolution of the galaxy luminosity function (LF) as a
function of spectral type using the Autofib redshift survey, a compendium of
over 1700 galaxy redshifts in various magnitude-limited samples spanning
b_J=11.5-24.0. To carry out this analysis we have developed a cross-correlation
technique which classifies faint galaxy spectra into one of six types based on
local galaxy templates. Tests and simulations show that this technique yields
classifications correct to within one type for more than 90% of the galaxies in
our sample. We have also developed extensions of the step-wise maximum
likelihood method and the STY parametric method for estimating LFs which are
applicable to recovering an evolving LF from multiple samples. We find that:
(i) The spectra and LF of E/S0 galaxies show no appreciable evolution out to at
least z ~ 0.5. (ii) Early-type spirals show modest evolution, characterised by
a gradual steepening of the faint end of their LF with redshift. (iii) Out to z
~ 0.5, the overall evolution of the galaxy population is dominated by changes
seen in late-type spirals. The characteristic luminosity (L^*) of these
galaxies appears to brighten with redshift and there are signs of strong
density evolution (a rapid increase in \phi^*). These effects appear to be
luminosity dependent so that the LF steepens at higher redshift. These trends
are accompanied by a steep increase in the median [OII] equivalent width,
implying a rapid increase in the star-formation rate with redshift at fixed
luminosity---a given star-formation rate is found at higher redshift in
galaxies of higher luminosity. We find that these conclusions are robust with
respect to spectral classification errors and the luminosity function
estimator.
