4: Isotropic Harmonizable Fields and Applications

We have treated so far various properties of harmonizable random processes and of their structural analyses. But some key applications show that we need to study also various aspects of some related problems if the index is not a linear set as time axis, but is multidimensional such as the one corresponding to the space-time problems in evolution, as in Physics and elsewhere. This, therefore, takes us to considerations of random fields whose index is just a directed set, as in space-time problems mentioned above. It was also found out in early analysis that a stationary field $\left\{X_t,t\in {ℝ}^n,n>1\right\}\subset L_0^2\left(P\right)$ which satisfies an isotropy condition is the trivial one (i.e., a constant field, with probability one). Such facts lead us to analyze and establish their nontrivial structured and related properties to be used, e.g., in isotropy for harmonizable fields.