Can any function on naturals be interpolated to a smooth function on reals? Let $f : \mathbb{N} \rightarrow \mathbb{N}$ be an arbitrary function from naturals to naturals. Is it always possible to find a function $g : \mathbb{R} \rightarrow \mathbb{R}$ such that

*

*for any $n \in \mathbb{N}$, we have $f(n) = g(n)$, and

*$g \in C^\infty$?

I'm asking because I was trying to prove a result about the ratios of functions from naturals to naturals and it occurred to me that if I could always interpolate to get back smooth functions from integers to integers, I could conceivably use l'Hopital's rule to resolve the limits.
Thanks!

Whittaker–Shannon interpolation, using the $sinc$ function, achieves that. http://en.wikipedia.org/wiki/Whittaker%E2%80%93Shannon_interpolation_formula
$$g(x)=\sum_{n=-\infty}^{+\infty}f_n\frac{\sin\pi(x-n)}{\pi(x-n)}.$$
