= Distribution annihilated by a function with simple zeros
If a smooth function $a$ on $\mathbb R$ has finitely many zeros $r_j$, all simple, then $aw=0$ exactly when $w=\sum_jc_j\delta_{r_j}$. Away from the zeros, divide a <test function> by the nonvanishing $a$ to show $w$ vanishes. Near $r_j$, write $a(x)=(x-r_j)b_j(x)$ with smooth nonzero $b_j$. The <kernel of multiplication by a coordinate>, after translation and multiplication by $b_j$, says that the local distribution is a multiple of $\delta_{r_j}$. A <partition of unity> gives the global sum. Simplicity of the zeros excludes derivatives of deltas.
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