Say you want to sample a signal $a(t)$ with a periodical delta function train from $-\infty$ to $+\infty$. Denote the Fourier transform of $a(t)$ as $A(\omega)$.
Consider two different approaches:
(1) directly sample $a(t)$ itself. The Fourier transform of the sampled result is then given by the sampling of $A(\omega)$ periodically.
(2) sample $a(t)\exp(-i\omega_0 t)$ and subtract the carrier $\exp(-i\omega_0 t)$ after sampling. The Fourier transform of this sampled result is given by periodically sampling $A(\omega+\omega_0)$ .
However, $A(\omega+\omega_0)$ and $A(\omega)$ does not (apparently) have a relation with each other, so there is no guarantee that the result of (1) and (2) are consistent.
What went wrong? I hoped I am not terrible negligent of something very basic.
My assumption that periodically sampling in time domain is completely characterized by periodically sampling in frequency is wrong.