The most common method is using Table. Valeriu used Map (/@) above, which also works. It is best not to begin user variables with upper case to prevent conflict with reserved names.
In[8]:= f[x_] := 1/2 (Cos[\[CapitalOmega] x] + 1);
(* using an iteration *)
g = Table[f[x], {x, 0, 10}]
Out[9]= {1, 1/2 (1 + Cos[\[CapitalOmega]]),
1/2 (1 + Cos[2 \[CapitalOmega]]), 1/2 (1 + Cos[3 \[CapitalOmega]]),
1/2 (1 + Cos[4 \[CapitalOmega]]), 1/2 (1 + Cos[5 \[CapitalOmega]]),
1/2 (1 + Cos[6 \[CapitalOmega]]), 1/2 (1 + Cos[7 \[CapitalOmega]]),
1/2 (1 + Cos[8 \[CapitalOmega]]), 1/2 (1 + Cos[9 \[CapitalOmega]]),
1/2 (1 + Cos[10 \[CapitalOmega]])}
(* using an explicit list *)
g = Table[f[x], {x, {0, 1, 2, 3, 4, 5}}]
Out[10]= {1, 1/2 (1 + Cos[\[CapitalOmega]]),
1/2 (1 + Cos[2 \[CapitalOmega]]), 1/2 (1 + Cos[3 \[CapitalOmega]]),
1/2 (1 + Cos[4 \[CapitalOmega]]), 1/2 (1 + Cos[5 \[CapitalOmega]])}