ABSTRACT
Often in applications, random fields X(t,x) which are functions of both space, and time, occur. It is convenient to write the parameter set as ( t ,x), where t IR, x IRn. These processes are often stationary in (t ,x) and stationary and isotropic in the spatial variable x. More specifically,
E(X(s, u)X(s + t, u + x)) = r(t, || x ||) where r is a function from IR2 to C. Yadrenko, [16] obtained the covariance of such a field as
r(s,t,x,y) = 2v n 2
- 0 eiw(s-t)Jv( ||x-y||)( || x-y ||)v d ( , ) (19)
(the notation as in (6).) In view of the motivation behind harmonizable fields, Swift [10] relaxed
the requirement of stationarity for these stationary spatially isotropic fields and gave the following definition
Definition 1. A random field X : IRk × IRn L20(P) is weakly harmonizable spatially isotropic if its covariance is expressible as
r(s,t,x,y) = 2v n2 IRk IRk 0 0 e
i •s-i '•t Jv(|| x – 'y||) || x – 'y ||v dF( , ,' , ')
(20) where F(•, •, •, •) is a function of bounded Fréchet variation.
