JH-singularity and JH-regularity of multivariate stationary processes over LCA groups

Abstract

Let G be an LCA group, I' its dual group, and H a closed subgroup of G such that its annihilator is countable. Let M denote a regular positive semidefinite matrix-valued Borel measure on I' and L^2(M) the corresponding Hilbert space of matrix-valued functions square-integrable with respect to M. For g ϵ G, let Z_g be the closure in L^2(M) of all matrix-valued trigonometric polynomials with frequencies from g+H. We describe those measures M for which Z_g = L^2(M) as well as those for which ∩ gϵG Zg = {0}. Interpreting M as a spectral measure of a multivariate wide sense stationary processon G and denoting by J_H the family of H-cosets we obtain conditions forJ_H-singularity and JH-regularity.