  We define a general scheme for the evolution of fragmentation functions which
resums both soft gluon logarithms and mass singularities in a consistent manner
and to any order, and requires no additional theoretical assumptions. Using the
Double Logarithmic Approximation and the known perturbative results for the
splitting functions, we present our scheme with the complete contribution from
the double logarithms, being the largest soft gluon logarithms. We show that
the resulting approximation is more complete than the Modified Leading
Logarithm Approximation even with the fixed order contribution calculated to
leading order only, and find, after using it to fit quark and gluon
fragmentation functions to experimental data, that this approximation in our
scheme gives a good description of the data from the largest x_p values to the
peak region in xi=ln (1/x_p), in contrast to other approximations. In addition,
we develop a treatment of hadron mass effects which gives additional
improvements at large xi.
