  We prove that there exists an extremal function to the Airy Strichartz
inequality, $e^{-t\partial_x^3}: L^2(\mathbb{R})\to L^8_{t,x}(\mathbb{R}^2)$ by
using the linear profile decomposition. Furthermore we show that, if $f$ is an
extremiser, then $f$ is extremely fast decaying in Fourier space and so $f$ can
be extended to be an entire function on the whole complex domain. The rapid
decay of the Fourier transform of extremisers is established with a bootstrap
argument which relies on a refined bilinear Airy Strichartz estimate and a
weighted Strichartz inequality.
