Recovery of square-integrable function from observations with gaussian errors

The aim of the article is to construct a solution for the problem of the optimal recovery (in the mean-square sense) of a measurable square-integrable (with respect to the Lebesgue measure) function defined on a finite-dimensional compact set. We prove optimal recovery procedure and establish conditions of its unbiasedness and consistency. Furthermore, an $\varepsilon^{\frac12}$-optimal stochastic recovery procedure is proposed and proved.