Representation of weakly harmonizable processes

Abstract

Weakly harmonizable processes are represented by a family of positive definite contractive linear operators in a Hilbert space. This generalizes the known result on weakly stationary processes involving a unitary family. A characterization of the vector Fourier integral of a measure on R → [unk], a reflexive space, is given, and this yields another characterization of weakly harmonizable processes when [unk] is a Hilbert space. Also these processes are shown to have associated spectra, yielding a positive solution to a problem of Rozanov.

• operator representation
• V-boundedness
• generalization of stationarity