Radial symmetric elements and the Bargmann transform
  We prove that a function or distribution on $\rr d$ is radial symmetric, if and only if its Bargmann transform is a composition by an entire function on $\mathbf C$ and the canonical quadratic function from $\cc d$ to $\mathbf C$. 
Marco Cappiello, Luigi Rodino, Joachim Toft
math.FA