Optimal recovery of a square integrable function from its observations with Gaussian errors

This paper is devoted to the mean-square optimal stochastic recovery of a square integrable function with respect to the Lebesgue measure defined on a finite-dimensional compact set. We justify an optimal recovery procedure for such a function observed at each point of its compact domain with Gaussian errors. The existence of the optimal stochastic recovery procedure as well as its unbiasedness and consistency are established. In addition, we propose and justify a near-optimal stochastic recovery procedure in order to: (i) estimate the dependence of the standard deviation on the number of orthogonal functions and the number of observations and (ii) find the number of orthogonal functions that minimizes the standard deviation.