A smooth enough, slow growing real-valued function $f(x)$, is uniquely determined by its values at integer arguments. I don't remember the name of the theorem and the conditions for this to be true. Can someone refresh my memory and provide the link / reference to this statement?
I worked on this topic a while back, and came up with the following formula, valid for a large class of even functions:
$$f(t) = \frac{\sin\pi t}{\pi}\cdot \Big[\frac{f(0)}{t} +\phi'(t)\sum_{k=1}^\infty (-1)^k \frac{f(k)}{\phi(t)-\phi(k)} \Big]$$ where (say) $\phi(t)=t^2$. I posted it on MO a while back, and even proved the formula. I remember the proof was quite simple, but I can't remember any of it. There's quite a bit of details in my earlier question from years ago, on MO, here. It was a question about interpolation.
I am again interested in this topic, and I applied the above formula to the real part of the Riemann zeta function $\zeta(t+i\sigma)$, and it works. It works at least for $0<t<20$ and $\sigma=0.8$. This surprises me a bit since that function does not satisfy the conditions required to guarantee convergence to the right solution (conditions that I found myself but they could be more strict than actually needed).
If you want to answer a question a bit more challenging than my initial question, you can answer whether my formula applies or not to the real part of the Riemann zeta function, even for large values of $t$ and different $\sigma$'s. But at this point I am only interested in a reference to the theorem in question.
Update
I wrote an article unrelated to this topic (it's about fuzzy spatial regression as a machine learning technique), and not yet completed. However Exercise 2 in section 5 deals with the problem described here. You can download the article here.
Figure: Dirichlet eta function (real part, bottom) and interpolation error (top)
