On the existence 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 nth order forward and backward differences of f at z₀ tend to zero as n→∞ whenever z₀ lies in the region of absolute convergence of f. Under the same hypothesis, we show that the Newton series of f centered at z₀ exists and converges in the half-plane (z)>(z₀). Assuming instead that f has a Lebesgue measurable inverse Fourier transform, we show that the nth order forward, backward, and central differences of f at any y are o(2ⁿ). Consequently, we show that the binomial sum ∑k≥0n kf(k) is o(2ⁿ).