Spline approximation of a random process with singularity

Let a continuous random process X   defined on [0,1] be (m+β)‐smooth(m+β)‐smooth, 0≤m,0<β≤10≤m,0<β≤1, in quadratic mean for all t>0t>0 and have an isolated singularity point at t=0. In addition, let X be locally like a m  -fold integrated β‐fractionalβ‐fractional Brownian motion for all nonsingular 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+β)n−(m+β) for the whole interval.