Spline approximation of a random process with singularity

Let a continuous random process X defined on [0,1] be ( m + β ) ‐smooth , 0 ≤ m , 0 < β ≤ 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 β ‐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 ) for the whole interval.