We consider the problem of estimating the operator norm of embeddings between certain weighted Paley-Wiener spaces. We discuss some qualitative properties for the extremal problems considered and provide some asymptotic results. For a few cases, we are able to to provide a precise formula for the sharp constant with techniques from the theory of reproducing kernel Hilbert spaces. As an application, these provide sharp constants to higher order Poincare inequalities via the Fourier transform.