Last night an old idea came back to me, an idea about a function where all the derivatives start from zero and then grow smoothly. I thought it would be impossible, but then I found some interesting stuff on Wikipedia. So, I learned to use SymPy and wasted a lot of time with it. Here’s a report of my (non-)findings.
(UPDATE: I did some numerical differentiation, which showed that h(x) does have negative derivatives. See details in this comment. A disappointment, although perhaps not a surprising one. It doesn’t however, necessarily mean the goal is impossible.)
So, if anyone knows whether such a function exists and what it looks like, please tell me.
You can think of the convolution as a process to smooth the function g by making its values at points around each t affect that at t. So, tau is the distance between t and another point, and Psi(tau) tells how much the other point contributes to the smoothing at point t. In a more decent situation, the integral in (7) would have been properly solved and tau would have disappeared, never to bother us again.