On the BSE and BED properties of the Beurling algebra $L^1(G,\omega )$

Let $G$ be a locally compact abelian group, and let $\omega :G\to [1,\mathrm{\infty })$ be a weight, i.e., $\omega$ is measurable, $\omega$ is locally bounded and $\omega (s+t)\le \omega (s)\omega (t)$ for all $s,t\in G$. If ${\omega }^{-1}$ is vanishing at infinity, then we show that the Beurling algebra $L^1(G,\omega )$ is both BSE- algebra and BED- algebra.