On the Form of Correlation Function for a Class of Nonstationary Field with a Zero Spectrum

Abstract

The present paper is devoted to the derivation of an explicit form of linearly representable random fields in the form h(x 1 ,x 2 ) = exp {i(x 1 A 1 + x 2 A 2 )} h , where h ∈ H, H is a Hilbert space, operators A 1 , A 2 are such that A 1 A 2 = A 2 A 1 and C 3 = 0 where C = A* 1 A 2 - A 2 A* 1 . The results obtained are the generalization of theorem proved by Livshits and Yantsevitch [4] and Yantsevich and Abbaui [6]. It is shown that a rank of nonstationary of field h(x 1 ,x 2 ) depends not only on a degree of nonself conjugation of A 1 , A 2 but on a degree of nilpotency of commutator C(C 3 = 0). In the present paper an explicit form of correlation function when the spectra of A 1 and A 2 lies in zero is derived.