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

TL;DR

This paper investigates conditions under which trigonometric polynomials with frequencies from a subgroup are dense in $L^eta$ spaces on dual groups of locally compact abelian groups, extending understanding of harmonic analysis.

Contribution

It provides necessary and sufficient conditions for the density of such polynomials in $L^eta$ spaces, generalizing previous results to broader group and measure settings.

Findings

Established criteria for density in $L^eta$ spaces.

Extended harmonic analysis to general LCA groups.

Connected subgroup frequency sets with approximation properties.

Abstract

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 $L^\alpha(\mu), \alpha \in (0,\infty)$.