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	UNIVERSITY OF WROCŁAW

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	WROCŁAW UNIVERSITY OF SCIENCE AND TECHNOLOGY

Contents of PMS, Vol. 41, Fasc. 1, pages 173 - 192
`DOI: 10.37190/0208-4147.41.1.11`
`Published online 22.4.2021`

$J_{H}$ -Singularity and $J_{H}$ -regularity of multivariate stationary processes over LCA groups Lutz Klotz Juan Miguel Medina Abstract: Let $G$ be an , $Γ$ its dual group, and $H$ a closed subgroup of $G$ such that its annihilator $Λ$ is countable. Let $M$ denote a regular Borel measure on $Γ$ 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 $⋂_{g \in G} Z_{g} = {0}$ . 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.
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