Good morning everybody. My question is as follows:

Let $K$ be a compact subset of $\mathbb R$ and let $v\in C^\infty(K)$.

Consider the finite difference operator $\Delta v(x)\doteq v(x+1)-v(x)$. It is known http://en.wikipedia.org/wiki/Indefinite_sum, that we can invert this operator $\Delta$ but the outcome is a family of functions which differ one from another by a periodic of modulo one function $p(x)=p(x+1)$.

It happens that in many cases e.g. if a Newton series exists that $p$ is actually a constant function.

Now the big issue here is to decide whether a Newton series exists which would be wonderful for my purposes. I am asking if this is ensured by the compactness of the set: to be more specific, we know that the Newton series holds for polynomials, and here maybe the result extends to smooth functions by some densities theorems like Stone Weierstrass or something along the same lines.

Is this result true? Can anybody provide me with a reference or a short proof if this is available?

I'm quite unused to these topics so any help is greatly appreciated.

Thanks again for the kindness and goodbye.