= Compact support solvability for a constant-coefficient ordinary differential equation
Use $D=-i\partial_x$. For a nonzero one-variable <polynomial> $P$ and $v\in\mathcal E'(\mathbb R)$, a compactly supported solution to $P(D)u=v$ exists exactly when $v$ annihilates every smooth solution of the transposed equation $P(-D)\varphi=0$. For simple roots $\alpha_j$, these test solutions span the exponentials $e^{-i\alpha_jx}$, so the condition is $\widehat v(\alpha_j)=0$. Dividing the entire transform by $P$ and applying <polynomial division preservation of exponential type> and the <Paley–Wiener–Schwartz theorem> constructs the compactly supported solution. It is unique because an entire function killed by a nonzero polynomial must vanish identically. For a root of multiplicity $r$, the conditions become $\widehat v^{(k)}(\alpha)=0$ for $0\leq k<r$, corresponding to $x^ke^{-i\alpha x}$.
Back to article page
