Building a ‘mollified’ transition function
Find a 
mapping ![\phi : [0,1] \rightarrow [a,b]](/cz/latexrender/pictures/edf877565d42579f481483d94a26f926.png "\phi : [0,1] \rightarrow [a,b]")
so that all orders of derivatives vanish at 0 and 1. I’m giving it a try…
Sad to say, that took me at least an hour. The integral 
of the mollifier function 
provides exactly what’s needed: take 
to be a suitably translated and scaled version of 
.
Possibly relevant posts:
- Dr. Singh’s Take on the Fourier Unicity Theorem (via Invariant Subspaces) (6/17/2005)
- Distribution function (1/14/2006)
- Complexification of the Rademacher comparison theorem? (8/18/2008)