  We study Hadamard operators on $S'(R^d)$ and give a complete
characterization. They have the form $L(S)=S*T$ where * here means the
multiplicative convolution and T is in the space of distributions which are
$\theta$-rapidly decreasing in infinity and at the coordinate hyperplanes. To
show this we study and characterize convolution operators on the space $Y(R^d)$
of exponentially decreasing $C^\infty$-functions on $R^d$. We use this and the
exponential transformation to characterize the Hadamard operators on $S'(Q)$,
$Q$ the positive quadrant, and this result we use as a building block for our
main result.
