Can any analytic function be represented as a sum of a Newton series and a periodic function?

Question

Suppose $f(x)$ is a real analytic function whose Newton series converges.

Conjecture.

$f(x)$ can be always represented as a sum of its Newton expansion and an 1-periodic function $g(x)$:

$$f(x)=g(x)+\sum_{k=0}^\infty \binom{x}k \Delta^k f\left (0\right)$$

Is this correct?

Answer 1

$\sin( 2 \pi x^2) $ is zero for $x \in \mathbb{Z}$, so has all $k^\text{th}$ differences zero, and is not a function with period $1$.