= Convolution of distributions with a compactly supported factor
{title2=$u*v$}
= Distributional convolution with a compact factor
{synonym}
If $v$ is a <compactly supported distribution> and $u$ is any <distribution>, define $\langle u*v,\phi\rangle=\langle u_x,\langle v_y,\phi(x+y)\rangle\rangle$. The inner function is smooth and supported in $\operatorname{supp}\phi-\operatorname{supp}v$, hence is a <test function> for $u$. Finite-order estimates prove continuity, while the <tensor product of distributions> proves commutativity after inserting compact cutoffs. For every <test function> $\psi$, $(u*v)*\psi=u*(v*\psi)$; evaluating at zero against reflected <test functions> proves uniqueness. If both <distributions> have <compact support>, $\operatorname{supp}(u*v)\subset\operatorname{supp}u+\operatorname{supp}v$ and the <convolution theorem> gives $\widehat{u*v}=\widehat u\widehat v$. Without a compact factor or another support/growth condition, arbitrary distributional <convolution> is not generally defined.
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