Let a continuous random process X defined on [0,1] be (m+β)-smooth,
0≤m,00 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 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+β) for the whole interval.Comment: 16 pages, 2 figure typos and references corrected, revised classes
definition, results unchange