Assume I have some function $t\to f(t)$ that is well behaved enough to have a Fourier Transform $w\to \mathcal F\{f(t)\}(w)$ as well as a power series expansion $$p(t) = \sum_{k\in \mathbb Z^+} c_kt^k$$ that converges everywhere. Furthermore assume that the Fourier Transform also has such a power series expansion. $$p_F(w) = \sum_{k\in \mathbb Z^+} d_kw^k$$

Can I derive some relationship between $\{c_k\}$ and $\{d_k\}$?

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  • Possibly useful: [(Question 7301)](https://math.stackexchange.com/questions/7301/connection-between-fourier-transform-and-taylor-series). – Jam May 17 '20 at 10:49
  • No it's not the same. This regards the monomial power series in the frequency variable not the trigonometric power series which is the transform itself. – mathreadler May 17 '20 at 15:20

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