Spectral representation of intrinsically stationary fields (Q713203)

The random fields are interpreted as Hilbert space-valued functions, and some results of \textit{Z. Sasvári} [in: V. Adamyan (ed.) et al., Modern analysis and applications. Basel: Birkhäuser. Operator Theory: Advances and Applications 190, 451--470 (2009; Zbl 1181.60054)] concerning the existence and representation of generalized correlation functions are studied in the slightly weaker notion of stationarizable processes. The notion of stationary field, which becomes intrinsically stationary when the considered Hilbert space-valued functions is finite dimensional, is introduced. Analogously to the notion of definitizable function in harmonic analysis, the notion of stationarizable random fields is considered. The aim of the paper is to obtain spectral representation for stationarizable and intrinsically stationary fields that generalize the known results for stationary fields. The study is done transferring the concept of process with weakly stationary increments to arbitrary locally compact Abelian groups.