The L2 structure of harmonizable random fields

In this paper, we aim to study a large class of harmonizable random fields (h.r.f., for short) when the indexing set is Zd, d ≥ 2 which contains several classes of stochastic processes. So, we present an appropriate spectral theory approach in order to characterize certain subclass of h.r.f. This class of processes is a natural generalization of the so-called wide sense stationary processes to random fields (r.f.) that have been investigated in many varieties of subjects and constitute an important class of nonstationary r.f. Moreover, since many phenomena across space in various sciences and industries can be interpreted as realizations of random fields, which may be further assumed to satisfy some invariance properties such as isotropy. So, the definition of isotropy is given and some examples in Rd are addressed. In the end, necessary and sufficient conditions ensuring the stability of the linear transformation of a particular class are studied.