On the convergence of Newton series and the asymptotics of finite differences

Suppose a complex function $f$ has a Lebesgue measurable inverse Laplace transform. We show that the $n$th order forward and backward differences of $f$ at $z_0\in \mathbb{C}$ tend to zero as $n\to \mathrm{\infty }$ whenever $z_0$ lies in the region of absolute convergence of $f$. Under the same hypothesis, we show that the Newton series of $f$ centered at $z_0$ exists and converges in the half-plane $\mathrm{\Re }(z)>\mathrm{\Re }(z_0)$. Assuming instead that $f$ has a Lebesgue measurable inverse Fourier transform, we show that the $n$th order forward, backward, and central differences of $f$ at any $y\in \mathbb{R}$ are $o(2^n)$. Consequently, we show that the binomial sum ${\sum }_{k\ge 0}\left(\genfrac{}{}{0px}{}{n}{k}\right)f(k)$ is $o(2^n)$.