Reconstruction of band limited processes from irregular samples
The main aim of this article is to study the path reconstruction or interpolation of certain stochastic processes (as opposed to the more commonly used approach of reconstruction on average). Justification for the preference of path over on average reconstruction is given together with brief literature reviews on the topics involved here: the regular sampling theorem, the theory of frames in complex separable Hilbert space, almost sure convergence in the sampling expansion of band-limited processes. Among the results on the reconstruction of processes is Theorem 3.4 which illustrates how previously known results of Belyaev, Piranashvili and Houdré can be generalised without the non-second-order irregular sampling frame work. Furthermore, the theorems of this paper generalise and extend the regular sampling results of Cambanis, Masry and Lee. Potential extensions of the material presented here to generalised random processes, sampling with derivatives, sampling for time-limited processes and second-order processes whose covariance function is tempered are not included in the current paper. Instead the article continues with a study of regular sampling with finitely many missing and/or finitely many irregular points. A large part of the story of this article is summarised as follows: ``... in a random environment and as far as sampling and reconstruction are concerned, a realization of a process is representative if the end points of the band are filtered out or if the spectrum of the process is smooth enough.'' The paper concludes with a study of the convergence rate in the interpolation formula and with a small section on truncation error analysis. None of the other types of errors occurring in an application of sampling series representations are discussed here.
scientific article; zbMATH DE number 796515