  We study boundary values of holomorphic functions in translation-invariant
distribution spaces of type $\mathcal{D}'_{E'_{\ast}}$. New edge of the wedge
theorems are obtained. The results are then applied to represent
$\mathcal{D}'_{E'_{\ast}}$ as a quotient space of holomorphic functions. We
also give representations of elements of $\mathcal{D}'_{E'_{\ast}}$ via the
heat kernel method. Our results cover as particular instances the cases of
boundary values, analytic representations, and heat kernel representations in
the context of the Schwartz spaces $\mathcal{D}'_{L^{p}}$, $\mathcal{B}'$, and
their weighted versions.
