Curve Fitting for a sequence

Question

Let $a_n\searrow 0$ be a strictly decreasing sequence. Is it always possible to find a $C^2$ or even $C^\infty$ function $f:[0,\infty)\to [0,\infty)$ so that $f$ is strictly decreasing and $f(n)=a_n$ for all $n$?

Could a cubic spline do that?

Answer 1

A $C_2$ cubic spline won't work. There are many examples of monotone data sequences for which the interpolating $C_2$ cubic spline is not monotone.

There are several ways to construct a monotone cubic spline (see here), but these curves are typically just $C_1$, not $C_2$.