  Let a continuous random process $X$ defined on $[0,1]$ be $(m+\beta)$-smooth,
$0\le m, 0<\beta\le 1$, in quadratic mean for all $t>0$ and have an isolated
singularity point at $t=0$. In addition, let $X$ be locally like a $m$-fold
integrated $\beta$-fractional Brownian motion for all non-singular points. We
consider approximation of $X$ by piecewise Hermite interpolation splines with
$n$ free knots (i.e., a sampling design, a mesh). The approximation performance
is measured by mean errors (e.g., integrated or maximal quadratic mean errors).
We construct a sequence of sampling designs with asymptotic approximation rate
$n^{-(m+\beta)}$ for the whole interval.
