  Under the standard assumptions on the variable exponent $p(x)$ (log- and
decay conditions), we give a characterization of the variable exponent Bessel
potential space $\mathfrak B^\alpha[L^{p(\cdot)}(\mathbb R^n)]$ in terms of the
rate of convergence of the Poisson semigroup $P_t$. We show that the existence
of the Riesz fractional derivative $\mathbb{D}^\al f$ in the space
$L^{p(\cdot)}(\rn)$ is equivalent to the existence of the limit
$\frac{1}{\ve^\al}(I-P_\ve)^\al f$. In the pre-limiting case $\sup_x
p(x)<\frac{n}{\al}$ we show that the Bessel potential space is characterized by
the condition $\|(I-P_\ve)^\al f\|_{p(\cdot)}\leqq C \ve^\al$
