There were many questions and answers about "The range of Fourier Transform on $L^1(\mathbb R)$ is dense in $C_0(\mathbb R)$. Using $C^2_c(\mathbb R)$ is the solution of it.
However, when it comes to $L^1(\mathbb T)$, which means $[-\pi,\pi)$(The equivalence class of $\mathbb R/{2n\pi}$, the range of Fourier Transform on $L^1(\mathbb T)$ is dense in $c_0(\mathbb Z)$. But, how could I prove it? I think I cannot use the similar way.. Could somebody give me an idea?