Spline approximation of a random process with singularity

Let a continuous random process X defined on [0,1] be (m+beta)-smooth, 0 <= m,0 < beta <= 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 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+beta)) for the whole interval. (C) 2010 Elsevier B.V. All rights reserved.