Convolution Theorem for the DTFT
This is sometimes called acyclic convolution to distinguish it from the cyclic convolution used for length sequences in the context of the DFT . Convolution is cyclic in the time domain for the DFT and FS cases (i.e., whenever the time domain has a finite length), and acyclic for the DTFT and FT cases.3.6
That is, convolution in the time domain corresponds to pointwise multiplication in the frequency domain.
Proof: The result follows immediately from interchanging the order of summations associated with the convolution and DTFT:
Correlation Theorem for the DTFT
Shift Theorem for the DTFT