Solution
= Solution
No. Non-entireness rules out a compactly supported solution, but the <Malgrange–Ehrenpreis theorem> gives a distributional <fundamental solution of a linear differential operator> $E$ with $P(D)E=\delta$. Since $v$ has compact support, the convolution $u=E*v$ is defined and satisfies
$$
P(D)u=(P(D)E)*v=v.
$$
One may choose a tempered fundamental solution for a constant-coefficient operator, so $u\in\mathcal S'(\mathbb R)$.