A Note on the Completeness of All Translates of a Function in the Orlicz Spaces

We give a characterization of those functions whose all translates are complete in certain Orlicz space $L^{\mathrm{\Phi }}(\mathbb{R})$. As a consequence, we identified those discrete sets $\mathrm{\Lambda }\subseteq \mathbb{R}$ such that there exists a function in $L^{\mathrm{\Phi }}(\mathbb{R})$ whose $\mathrm{\Lambda }$-translates are complete. We then prove the completeness of all translates of any simple step function in other Orlicz spaces.