Paley–Wiener–Schwartz theorem

The Fourier transform of a distribution supported in a compact set $K$ extends to an entire function bounded by a polynomial times $e^{H_K(\mathrm{Im}z)}$, where $H_K$ is the supporting function of $K$. Conversely, every entire function with such a bound is the Fourier--Laplace transform of a distribution supported in $K$.