What I want. I write an operator expression, and Mathematica performs it on a function.
Examples:
E^(b D): f[x] -> f[x+b]
a^(x_ D): f[x] -> f[a x]
E^(Pi I /4 (D^2 - x_^2 +1)) : Fourier transform (unitary, angular velocity)
E^(D^2) : Weierstrass transform
etc. See here more: https://math.stackexchange.com/a/1495596/2513
