Linear interpolation, extrapolation, and prediction of random space-time fields with a limited domain of measurement

Formulas are derived for linear (least-square) reconstruction of multidimensional ({em e.g.}, space-time) random fields from sample measurements taken over a limited region of observation. Data may or may not be contaminated with additive noise, and the sampling points may or may not be constrained to lie on a periodic The solution of the optimum filter problem in wave-number space is possible under certain restrictive conditions: 1) that the sampling locations be periodic and occupy a sector of the Euclidean sampling space, and 2) that the wave-number spectrum be factorable into two components, one of which represents a function nonzero only within the data space, the other only within the sector imaging the data space through the origin. If the values of the continuous field are accessible before sampling, a prefiltering operation can, in general, reduce the subsequent error of reconstruction. However, the determination of the optimum filter functions is exceedingly difficult, except under very special circumstances. A one-dimensional second-order Butterworth process is used to illustrate the effects of various postulated constraints on the sampling and filtering configuration.