Some Classes of Random Fields inn-Dimensional Space, Related to Stationary Random Processes

Abstract

The purpose of this paper is to establish a spectral theory for certain types of random fields and random generalized fields (multidimensional random distributions) in the Euclidean n-space $R_n $ similar to the well-known spectral theory for stationary random processes.Let D denote the Schwartz space of all complex-valued $C_\infty $ functions $\varphi ({\bf x})$ defined on $R_n $ whose carrier is compact. Following Ito [6] and Gelfand [7] we shall call the random linear functional $\xi (\varphi )$ on D satisfying (1.4) the random generalized field. We can identify a continuous random field $\xi ({\bf x})$ on $R_n $ with a random generalized field (1.5) and therefore we can consider the ordinary random fields $\xi ({\bf x})$ as special cases of random generalized fields.We shall only deal with the first moment $m(\varphi )$ and the second moment $B(\varphi _1 ,\varphi _2 )$ of the random generalized field $\xi (\varphi )$ and shall call them a mean value functional and a covariance functional of this fie...