Optimal designs for approximating the path of a stochastic process

We consider a centered stochastic process {X(t):t epsilon T} with known and continuous covariance function. On the basis of observations X(t(1)),...,X(t(n)) we approximate the whole path by orthogonal projection and measure the performance of the chosen design d = (t(1),..., t(n))' by the corresponding mean squared L(2)-distance. For covariance functions on T-2 = [0, 1](2), which satisfy a generalized Sacks-Ylvisaker regularity condition of order zero, we construct asymptotically optimal sequences of designs. Moreover, we characterize the achievement of a lower error bound, given by Micchelli and Wahba (1981), and study the question of whether this bound can be attained.