On recovering missing values with minimal error

The paper studies frequency criterions of recoverability of a single missing value from an observed sequence. Stochastic Gaussian stationary processes that are non recoverable in this sense are called minimal processes; they have a degenerate spectral density. The paper considers this problem in a pathwise setting without probabilistic assumptions; recoverability criterions are formulated in the terms of a degeneracy of the Z-transforms. For processes with non-degenerate Z-transform, an optimal recovering algorithm is suggested. This algorithm is based on projections on a closest recoverable sequence; it leads to explicit formulae for the recovered value. Some robustness of the solution with respect to noise contamination and truncation is established.