Characterization of the spectra of periodically correlated processes (Q5943591)

The main result proves that a tempered distribution \(F\) on \(R^{2}\) is the spectrum of a periodically correlated process with period \(T>0\) if and only if \(F\) is positive definite and \(F = \sum_{k} F_{k}\), where, for each integer \(k\), \(F_{k}\) is a uniformly bounded complex measure on \(R^{2}\) supported on the line \(L_{k}= \{ (s,s + 2 \pi k)/T: s \in R\}\). The form of each \(F_{k}\) is also derived. Some previous work of the author [Stud. Math. 136, No. 1, 71-86 (1999; Zbl 0947.60032)] plays a key role in the proof.