Show that f is a polynomial

Suppose $f$ is an entire function on $\mathbb{C}^n$ that satisfies for every $\epsilon>0$ a growth-condition $$|f(z)|\leq C_{\epsilon}(1+|z|)^{N_{\epsilon}}e^{\epsilon | \text{Im}\,z|}$$

Show that $f$ is a polynomial. (Hint: study $\hat{f} = \mathcal{F}(f)$ the Fourier-transform).

I know I'm supposed to apply the Paley-Wiener-Schwartz Theorem, but not sure how.;.

(See http://en.wikipedia.org/wiki/Paley%E2%80%93Wiener_theorem below).

Any suggestions and/or tips are greatly appreciated. Thnx.

Well, I think it should be clear as suggested by Aaron that the Schwartz's Theorem in your link implies that for every $ \epsilon >0 $ you have a distribution $ v $ supported in the closed ball $ B(0,\epsilon) $, and $ f = \hat{v} $. So we have the distribution $v$ with $ f = \hat{v} $ and $ supp(v) = \{0\} $. Hence it is a well known result that for some $k $ $$ v = \sum_{|\alpha | = k}C_\alpha D^\alpha \delta $$ Hence $f(\xi) = \hat{v}(\xi) = \sum_{|\alpha | = k}C_\alpha \hat{(D^\alpha \delta)}(\xi)= \sum_{|\alpha | = k}C_\alpha (2\pi i \xi)^\alpha $ is indeed a polynomial.