  The transfer function $T(k)$ of dark matter (DM) perturbations during matter
domination is obtained by solving the collisionless Boltzmann-Vlasov equation.
We find an \emph{exact} expression for $T(k)$ for \emph{arbitrary} distribution
functions of decoupled particles and initial conditions}. We find a remarkably
accurate and simple approximation valid on all scales of cosmological relevance
for structure formation in the linear regime. The natural scale of suppression
is the free streaming wavevector at matter-radiation equality, $ k_{fs}(t_{eq})
= [{4\pi\rho_{0M}}/{[< \vec{V}^2> (1+z_{eq})]} ]^\frac12 $. An important
ingredient is a non-local kernel determined by the distribution functions of
the decoupled particles which describes the \emph{memory of the initial
conditions and gravitational clustering} and yields a correction to the fluid
description. Distribution functions that favor the small momentum region lead
to an \emph{enhancement of power at small scales} $ k > k_{fs}(t_{eq}) $. For
DM thermal relics that decoupled while ultrarelativistic we find $
k_{fs}(t_{eq}) \simeq 0.003 (g_d/2)^\frac13 (m/\mathrm{keV})
[\mathrm{kpc}]^{-1} $, where $ g_d $ is the number of degrees of freedom at
decoupling. For WIMPS we obtain $ k_{fs}(t_{eq}) = 5.88 (g_d/2)^\frac13 (m/100
\mathrm{GeV})^\frac12 (T_d/10 \mathrm{MeV})^\frac12 [\mathrm{pc}]^{-1} $. For
$k\ll k_{fs}(t_{eq})$, $T(k) \sim 1-\mathrm{C}[k/k_{fs}(t_{eq})]^2 $ where $C
=\mathrm{O}(1)$ for all cases considered and simple and accurate fits for
\emph{small} scales.
