  We study the perturbative structure of threshold enhanced logarithms in the
coefficient functions of deep inelastic scattering (DIS) and semi-inclusive
$e^+e^-$ annihilation (SIA) processes and setup a framework to sum them up to
all orders in perturbation theory. Threshold logarithms show up as the
distributions $((1-z)^{-1} \log^i(1-z))_+$ from the soft plus virtual (SV) and
as logarithms $\log^i(1-z)$ from next to SV (NSV) contributions. We use the
Sudakov differential and the renormalisation group equations along with the
factorisation properties of parton level cross sections to obtain the resummed
result which predicts SV as well as next to SV contributions to all orders in
strong coupling constant. In Mellin $N$ space, we resum the large logarithms of
the form $\log^i(N)$ keeping $1/N$ corrections. In particular, the towers of
logarithms, each of the form $a_s^n/N^\alpha \log^{2n-\alpha} (N),
a_s^n/N^\alpha \log^{2n-1-\alpha}(N) \cdots $ etc for $\alpha =0,1$, are summed
to all orders in $a_s$.
