Abstract: Let $G$ be an , $\mathrm{\Gamma }$ its dual group, and $H$ a closed subgroup of $G$ such that its annihilator $\mathrm{\Lambda }$ is countable. Let $M$ denote a regular Borel measure on $\mathrm{\Gamma }$ and $L^2(M)$ the corresponding Hilbert space of functions square-integrable with respect to $M$. For $g\in G$, let $Z_g$ be the closure in $L^2(M)$ of all 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 $\bigcap _{g\in G}{Z}_g=\{0\}$. Interpreting $M$ as a spectral measure of a multivariate wide sense stationary process on $G$ and denoting by $J_H$ the family of, $H$ {cosets}, we obtain conditions for $J_H$ -singularity and $J_H$ - regularity.

2010 AMS Mathematics Subject Classification: Primary 42A10, 43A25, 60G25, 43A05; Secondary 94A20.

Keywords and phrases: LCA group, multivariate stationary process, positive semidefinite matrix-valued measure, trigonometric approximation, $J_H$- singularity, $J_H$-regularity, sampling.