Exponential of a polynomial of the differential operator

Question

Given that

$$\exp(aD)f(x)=f(x+a)$$

where $\exp(D)$ is the exponential of the differential operator $D$, is there a similar closed-form, general expression for $\exp(g(D))f(x)$, where $g(D)$ is a polynomial function of $D$?

Answer 1

In general if $a$ is a very nice function, not necessarily a pollinomial, then $$ a(D)f(x) = \int _{-\infty} ^\infty a(\xi) \hat f (\xi) e ^{2 \pi i \xi x } d\xi$$ Sadly there is no closed form for most $a$, that formula is a consequence of the fact that the Fourier transform satisfies $FD=FQ$ where $Q f(x)= x f(x)$.

There was a post with some nice examples I'm going to look for it