Sampling theorems and bases in a Hilbert space

Abstract

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