Approximations and representations for Fourier transforms

Abstract:

$G$ is a locally compact abelian group with dual $\Gamma$. If $p(\gamma ) = \sum \nolimits _1^N {{a_n}({x_n},\gamma )}$ is a trigonometric polynomial, its capacity, by definition is $\Sigma |{a_n}|$. The main theorem is: Let $\varphi$ be a measurable function defined on the measurable subset $\Lambda$ of $\Gamma$. If $\varphi$ can be approximated on finite sets in $\Lambda$ by trigonometric polynomials of capacity at most $C$ (constant), then $\varphi = \hat \mu$, locally almost everywhere on $\Lambda$, where $\mu$ is a regular bounded measure on $G$ and $||\mu || \leqq C$.