Sampling Theorems and Bases in a Hilbert Space

A unified approach to sampling theorems for (wide sense) stat ionary random processes rests upon Hilbert space concepts. New results in sampling theory are obtained along the following lines: recovery of the process x(t) from nonperiodic samples, or when any finite number of samples are deleted; conditions for obtaining x (t) when only the past is sampled; a criterion for restoring x(t) from a finite number of consecutive samples; and a minimum mean square error est imate of x(t) based on any (possibly nonperiodie) set of samples. In each case, the proofs apply not only to the recovery of x(t), but are extended to show that (almost) arbitrary linear operations on x (t) can be reproduced by linear combinations of the samples. Further generality is at tained by use of the spectral distr ibution function F(. ) of x(t), without assuming F( . ) absolutely continuous.