Spline approximation of random processes and design problems

We consider the spline approximation of a continuous (continuously diierentiable) random process with nite second moments based on n observations of the process (and its derivatives). The performance of the approximation is measured by mean errors (e.g., integrated or maximal quadratic mean errors). For Hermite interpolation splines, an optimal rule sets n observation locations (i.e., a design, a mesh). While, for a xed n, an optimal rule is diicult to construct, we nd the sequence of designs with asymptotically optimal properties as n ! 1. We investigate the class of locally stationary random processes whose local behavior is like m-fold integrated fractional Brownian motion for a given non-negative m in the quadratic mean sense.