Density of spaces of trigonometric polynomials with frequencies from a subgroup in $L^{\alpha }$-spaces

Abstract

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