In the desktop world they make a snapshot of the desktop then make it the background for drawing your application's images. The problem with this approach is when you try to move the application window (in which case the background will most probably turn out wrong).
But this sort of problem can be fixed by adding some window drag/move listeners.
You should be able to apply this concept to applets I suppose.
Anyway, it looks like I found an example of what you actually need.