isomorphism between $l^{2}(Z)$and $L^{2}(S,C)$

Question

i want to prove that $l^{2}(Z)$is isometric to $L^{2}(S,C)$, $S=\{z\in C,|z|=1\}$ may be the answer is to take an operator A such that :$Af=c_{n}$ where $c_{n}$ is Fourier coefficient of f ? is is true if not please i need hint,thanks