A Characterisation of the Fourier transform on the Schwartz-Bruhat space of locally compact Abelian groups

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.