  It is common knowledge that the Fourier transform enjoys the convolution
property, i.e., it turns convolution in the time domain into multiplication in
the frequency domain. It is probably less known that this property
characterizes the Fourier transform amongst all linear and bounded operators
$T:L^1 \longrightarrow C^b.$ Thus, a natural question arises: are there other
features characterizing Fourier transform besides the convolution property? We
answer this query in the affirmative by investigating the time differentiation
property and its discrete counterpart, used to characterize discrete-time
Fourier transform. Next, we move on to locally compact abelian groups, where
differentiation becomes meaningless, but the Fourier transform can be
characterized via time shifts. The penultimate section of the paper returns to
the convolution characterization, this time in the context of compact (not
necessarily abelian) groups. We demonstrate that the proof existing in the
literature can be greatly simplified. Lastly, we hint at the possibility of
other transforms being characterized by their properties and demonstrate that
the Hankel transform may be characterized by a Bessel-type differential
property.
