Density of spaces of trigonometric polynomials with frequencies from a subgroup in L^α-spaces

Let $G$ be an LCA group, $H$ a closed subgroup, $\varGamma$ the dual group of $G$ and $\mu$ be a regular finite non-negative Borel measure on $\varGamma$. We give some necessary and sufficient conditions for the density of the set of trigonometric polynomials on $\varGamma$ with frequencies from $H$ in the space $L^\alpha(\mu), \alpha \in (0,\infty)$.