Given that $G$ is the Green's function of some $m^{\rm th}$ order ODE with homogeneous Cauchy conditions, what does then $$ \frac{d^n G}{d t^n}\bigg|_{t = 0} $$ mean for $n > m$? When $n < m$, $G^{(n)}(0) = 0$ by the definition.
As far as I understand, in general, $G$ is a distribution, so that $G^{(n)}$ is a distribution as well. I am confused about what sense could $G^{(n)}(0)$ have and how to evaluate it?
Edit
Forgot to mention that in my particular problem $G(t) = \theta(t) w_0(t)$, where $\theta$ is the Heaviside function, $w_0$ is a proper function.