  Alesker has proved the existence of a remarkable isomorphism of the space of
translation-invariant smooth valuations that has the same functorial properties
as the classical Fourier transform. In this paper, we show how to directly
describe this isomorphism in terms of the Fourier transform on functions. As a
consequence, we obtain simple proofs of the main properties of the
Alesker--Fourier transform. One of these properties was previously only
conjectured by Alesker and is proved here for the first time.
