  We consider the rate of piecewise constant approximation to a locally
stationary process $X(t),t\in [0,1]$, having a variable smoothness index
$\alpha(t)$. Assuming that $\alpha(\cdot)$ attains its unique minimum at zero
and satisfies the regularity condition, we propose a method for construction of
observation points (composite dilated design) and find an asymptotics for the
integrated mean square error, where a piecewise constant approximation $X_n$ is
based on $N(n)\sim n$ observations of $X$. Further, we prove that the suggested
approximation rate is optimal, and then show how to find an optimal constant.
