Polar Spectral Representation of a Homogeneous and Isotropic Random Field

Abstract

Some representations of the random field, a generalization or the usual random process in multi-dimensional space, are given. The primary object is to obtain a new polar spectral representation for the homogeneous and isotropic random field applicable to a definite physical problem. The argument starts with constructing the moving average in multi-dimensional space by defining a Wiener integral in multi-dimensional space. The Fourier decomposition of the square integrable function specifying the moving average is shown to induce the ordinary spectral decomposition of the random field. Similary a polar spectral representation of a homogeneous and isotropic random field is derived explicitly for 2-and 3-dimensional cases in terms of countably infinite numbers of mutually independent random spectral measures. The correlation function and the spectral density are related to each other through a Hankel transformation. It is shown that the polar spectral representation takes a very simple form when a random pha...