Existence of smoothed stationary processes on an interval

The paper identifies the class of stationary processes on an interval which share a given stationary Gaussian process as kth derivative. The membership requirement involves the norm in the reproducing kernel space associated with the process sought as derivative. Some explicit results are obtained when working with the Ornstein-Uhlenbeck and the linear kernels. These latter facts are useful in an adaptive Bayesian modelling of computer experiments; some remarks are given about this type of analysis.