Стохастическое восстановление квадратично интегрируемых функций
The article is devoted to the solution of stochastic recovery problem for a square-integrable (with respect to the Lebesgue measure) functions defined on the real number line from observations with additive white Gaussian noise for the case of discrete time. This is the problem of nonparametric (infinitedimensional) estimation. We prove optimality of the suggested recovery procedure (in the mean-square sense) with respect to the product of the Lebesgue measure and the Gaussian measure. Recovery algorithm for such square-integrable functions is provided. It is proved, that constructed nonparametric recovery procedure for a square-integrable function provides unbiased and consistent recovery for an unknown function. This new result has not been published yet. Furthermore, we suggest and prove almostoptimality of recovery procedure for a smooth recovered functions which gives non-improvable (in the
sense of magnitude order) estimation of how number of orthogonal functions depends on observations' number. That is important, the error of constructed almost-optimal recovery procedure in comparison with optimal recovery procedure is no more than fifty per cent.