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We consider a stationary spatio-temporal random process and assume that wehave a sample. By defining a sequence of discrete Fourier transforms atcanonical frequencies at each location, and using these complex valued randomvarables as observed sample, we obtain expressions for the spatio-temporalcovariance functions and the spectral density functions of the spatio-temporalrandom processes. These spectra correspond to non separable class of randomprocesses. The spatio-temporal covariance functions, obtained here arefunctions of the spatial distance and the temporal frequency and are similar toMatern class. These are in terms of modified Bessel functions of the secondkind. and the parameters are in terms of the second order spectral densityfunctions of the random proces and the spatial distances. We consider theestimation of the parameters of the covariance function and also brieflymention their asymptotic properties. The estimation of the entire data at aknown location, and also the estimation of a value given the above sample isalso considered. The predictors are obtained using the vectors of DiscreteFourier Transforms. We also describe a statistical test for testing theindependence of the m spatial time series (testing for spatial independence)using the Finite Fourier Transforms and it is based on the likelihood ratiotest of complex valued random variables The methods are illustrated with realdata.  Keywords: Discrete Fourier Transforms, Covariance functions, Spectral densityfunctions, Space-Time Processses, Prediction(kriging) Laplacian operators,Frequency Variogram, Tests for independence, Whittle likelihood.

A space-time covariance function for spatio-temporal random processes and spatio-temporal prediction (kriging)