torben@arizona.UUCP (12/13/83)
What is this apart from a restatement of Taylor's Theorem? Of course, your assumptions must be modified. For a function f which is just continuously differentiable, your result won't hold. You need to have C infinity.
fulk@sunybcs.UUCP (Mark Fulk) (12/14/83)
To prove exp(D) f = f, you need more than differentiability OR even C infinity; you need that f is analytic. A good example: f(x) = exp(-1/x*x) if x>0; f(x) = 0 otherwise. Taking x in the statement of the problem to be 0, we would have: (exp(D) f)(0) = 0 != f(1) = 1/e. This is, of course, a standard example from advanced calculus. -- Mark Fulk Department of Computer Science fulk.buffalo@udel-relay SUNY at Buffalo ...!rocksvax!sunybcs!fulk 4226 Ridge Lea Rd. (716)831-3061 Amherst, NY 14226