Abstract: Let \(G\) be an , \(\Gamma\) its dual group, and \(H\) a closed subgroup of \(G\) such that its annihilator \(\Lambda\) is countable. Let \(M\) denote a regular Borel measure on \(\Gamma\) 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 \(\bigcap_{g\in G} Z_g=\set{0}\). 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.