Spline approximation of a random process with singularity

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.