Ditkin sets for some functional spaces and applications to grand Lebesgue spaces

Let $G$ be a locally compact Abelian group with dual group $\widehat{G}$ and Haar measures $d\mu$ and $\widehat{d}\mu$ respectively. In this work we have proved that if $X$ is an essential Banach ideal in Beurling algebra $L_{\omega }^1(G),$ then a closed subset $E\subset \widehat{G}$ is a Ditkin set for $X$ if and only if $E$ is a Ditkin set for $L_{\omega }^1(G)\mathrm{.}$ Next, as an application we have investigated the Ditkin sets for grand Lebesgue space $L^{p),\theta }(G)$ and Ditkin sets for $[L^p(G){]}_{L^{p),\theta }}$, where $[L^p(G){]}_{L^{p),\theta }}$ is the closure of the set $C_c(G)$ in $L^{p),\theta }(G)$.