The sampling-reconstruction procedure with a limited number of samples of stochastic processes and fields on the basis of the conditional mean rule

The known classical conditional mean rule is applied in order to obtain the statistical description of Sampling-Reconstruction Procedure (SRP) of stochastic processes and fields when the number of samples is limited and arbitrary. As a result the principal statistical SRP characteristics are obtained: the optimal reconstruction function (or the set of optimal basic functions) and the minimum error reconstruction functions. Another proof of Balakrishnan's reconstruction procedure is presented here. We give some comments of Balakrishnan's theorem also. The optimal SRP principal characteristics of several sampling types of the Gaussian fields are obtained. The SRP of Markov non-Gaussian continuous processes is described also. We demonstrate that the optimal reconstruction function of non-Gaussian processes is generally non-linear function of samples. All considered examples have the applied meaning and they can be productively used as the practical recommendations in order to choose some principal parameters of SRP of random processes and fields of different types.