  In this note, we study the general form of a multiplicative bijection on
several families of functions defined on manifolds, both real or complex
valued. In the real case, we prove that it is essentially defined by a
composition with a diffeomorphism of the underlying manifold (with a bit more
freedom in families of continuous functions). Our results in the real case are
mostly simple extensions of known theorems. We then show that in the complex
case, the only additional freedom allowed is complex conjugation. Finally, we
apply those results to characterize the Fourier transform between certain
function spaces.
