My (naive) question is whether it is possible to take the Fourier transform of a Taylor series?
Could one use multiplication with $\delta$ to get the function sampled at the point of expansion and the derivative theorem to unpackage the derivatives?
Is this approach inherently flawed? Does it work under certain assumptions?
Intuition suggests that since the Taylor approximation is localized in the time domain it should be spread out in the frequency domain, but it would be nice to see this expressed...