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 $\{d_k\}$?