  We show that if $u$ is a compactly supported distribution on the complex
plane such that, for every pair of entire functions $f,g$, \[ \langle
u,f\overline{g}\rangle=\langle u,f\rangle\langle u,\overline{g}\rangle, \] then
$u$ is supported at a single point. As an application, we complete the
classification of all weighted Dirichlet spaces on the unit disk that are de
Branges-Rovnyak spaces by showing that, for such spaces, the weight is
necessarily a superharmonic function.
