  We define the local trace function for subspaces of $\ltworn$ which are
invariant under integer translation. Our trace function contains the dimension
function and the spectral function defined by Bownik and Rzeszotnik and
completely characterizes the given translation invariant subspace. It has
properties such as positivity, additivity, monotony and some form of
continuity. It behaves nicely under dilations and modulations. We use the local
trace function to deduce, using short and simple arguments, some fundamental
facts about wavelets such as the characterizing equations, the equality between
the dimension function and the multiplicity function and some new relations
between scaling functions and wavelets.
