  Let $V$ be a finite dimensional real vector space. In the article we
construct an isomorphism between the space of smooth translation invariant
valuations on convex subsets of $V$ and the space of such valuations (twisted
by densities) on the dual space $V^*$. This isomorphism commutes with the
natural action of the full linear group on both spaces of valuations, maps
product on the source (introduced in arXiv:math/0301148) to the convolution on
the target (introduced in arXiv:math/0607449), and satisfies a Plancherel type
inversion formula. As an application, a version of the hard Lefschetz theorem
for valuations is proved.
