I just learned about the dirac comb
$$ Ш_T(t) = \sum_n \delta(t-nT) $$
and wanted to use it and the convolution theorem to understand the spectrum of a train of sinc pulses $$ x(t) = \text{sinc}(t)* Ш_1(t) $$
However, my considerations for the fourier transformed lead to an apparently incorrect result:
The convolution theorem states that the convolution in the time domain becomes a multiplication in the frequency domain. The fourier transformed of $\text{sinc}$ becomes a $\text{rect}$, while the fourier transformed of the dirac comb is a dirac comb again. In short, periodizing in the time domain samples in the frequency domain:
$$ X(f) = \text{rect}(f) \cdot Ш_1(f) $$
This result does not make sense to me however, as I would interpret it as a single dirac delta $\delta(f)$, since all other dirac deltas fall outside the $\text{rect}$-function and become zero. But a single $\delta(f)$ is also the fourier transformed of a single oscillation, so two different function would have the same fourier transformed.
Where am I making a mistake?