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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...

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See rule 308 on the wikipedia page for the Fourier transform: en.wikipedia.org/wiki/Fourier_transform. If you taylor expand in the time domain, you have polynomials $t^n$ which are "spread out" so if anything localization occurs in the frequency domain. Also check out the wikipedia page on the Dirac delta function for the definition of the distributional derivative that appears in the FT of polynomials. –  Graham Hesketh Jun 26 '13 at 16:11
    
But this is a formal definition, in practice it is often useless because Taylor series often only converge over a restricted domain and the FT evaluates the function over the whole of the real line. The obvious exception being entire functions equal to there Taylor series everywhere. –  Graham Hesketh Jun 26 '13 at 16:17

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