  We give a complete characterisation of the spaces $\dot{B}^{\alpha}_{p,q}$
and $\dot{F}^{\alpha}_{p,q}$ by using a non-smooth kernel satisfying near
minimal conditions. The tools used include a Stromberg-Torchinsky type estimate
for certain maximal functions and the concept of a distribution of finite
growth, inspired by Stein. Moreover, our exposition also makes essential use of
a number of refinements of the well-known Calderon reproducing formula. The
results are then applied to obtain the characterisation of these spaces via a
fractional derivative of the Poisson kernel.
