In this question about discrete-analytic functions (that is functions, who equal to their Newton series) I asked for a solution for the following problem:
Is there a method to extend the notion of discrete analiticity to a range of functions for which Newton series does not converge
I received no answer, but now I have a promising formula at hand.
$$f(x)=\frac{1}{2\pi}\int_{-\infty}^{+\infty} (- i \omega)^x \int_{-\infty}^{+\infty}e^{i\omega t}\sum_{m=0}^\infty \frac{f(m)t^m}{m!}dt \, d\omega$$
Code in Mathematica:
Nf[x_] := InverseFourierTransform[FourierTransform[GeneratingFunction[f[n]/n!, n, t], t, w] (- I w)^x, w, 0]
It seems this transform should converge for some cases where the Newton's series does not, for instance, for the function $\sin \frac{\pi x}2$ for which the Newton's series does not converge.
I wonder whether the proposed formula a viable generalization for Newton's series and on what set of functions does it converge? Does it substantially extend the area of convergence of Newton's expansion?
Does it always produce the same result as the Newton's series?
$$f(x) = \sum_{k=0}^\infty \binom{x}k \Delta^k [f]\left (0\right)$$
