A remark on the Strichartz inequality in one dimension

In this paper, we study the extremal problem for the Strichartz inequality for the Schr"{o}dinger equation on ${\mathbb{R}}^2$. We show that the solutions to the associated Euler-Lagrange equation are exponentially decaying in the Fourier space and thus can be extended to be complex analytic. Consequently we provide a new proof to the characterization of the extremal functions: the only extremals are Gaussian functions, which was investigated previously by Foschi and Hundertmark-Zharnitsky.