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 χ( t ) from nonperiodic samples, or when any finite number of samples are deleted; conditions for obtaining χ( t ) when only the past is sampled; a criterion for restoring χ( t ) from a finite number of consecutive samples; and a minimum mean square error estimate of χ( t ) based on any (possibly nonperiodic) set of samples. In each case, the proofs apply not only to the recovery of χ( t ), but are extended to show that (almost) arbitrary linear operations on χ( t ) can be reproduced by linear combinations of the samples. Further generality is attained by use of the spectral distribution function F (·) of χ( t ), without assuming F (·) absolutely continuous.