= Smoothing convolution with a test function
{title2=$E*f$}
For any <distribution> $E\in\mathcal D'$ and any <test function> $f$, the <convolution> $(E*f)(x)=\langle E_y,f(x-y)\rangle$ is a <smooth function>. On a compact set of $x$ values, the translated test functions and all their derivatives have supports in one fixed compact set, so distributional continuity permits every derivative:
$$
\partial^\alpha(E*f)(x)=\langle E_y,\partial^\alpha f(x-y)\rangle.
$$
For constant-coefficient operators, $P(D)(E*f)=(P(D)E)*f$. Hence a <fundamental solution of a linear differential operator> supplies a smooth particular solution for every compactly supported smooth datum. No temperedness of $E$ is needed.
Back to article page
