  We obtain a characterisation of the Fourier transform on the space of
Schwartz-Bruhat functions on locally compact Abelian groups. The result states
that any appropriately additive bijection of the Schwartz space onto itself,
which interchanges convolution and pointwise products is essentially the
Fourier transform. The proof of this result is very similar to that obtained by
the author recently for the Euclidean Fourier transform.
