The zeros of point evaluation extremals in Paley-Wiener spaces

For $1\le p<\mathrm{\infty }$, ${\phi }_p$ is the function of minimal $L^p$ norm among all functions $f$ of exponential type at most $\pi$ for which $f(0)=1$. We start from the ansatz that the positive zeros ${\tau }_n$ of ${\phi }_p$ are close to $n+1\mathrm{/}p-1\mathrm{/}2$ and use a weighted convex correction to first place them in an ${\mathrm{\ell }}^2$-neighborhood of this sequence. A bootstrap argument then improves this to a {Poincaré asymptotic} expansion of ${\tau }_n-(n+\frac{1}{p}-\frac{1}{2})$ in terms of odd powers of $(n+\frac{1}{p}-\frac{1}{2}{)}^{-1}$. We also get precise information about the Fourier transform ${\widehat{\phi }}_p$, a result that in particular implies that for all integers $n\ge 2$, ${\widehat{\phi }}_{2n}$ is an integrable function which, when convolved with itself $2n-2$ times, is constant on the interval $[-\pi ,\pi ]$. In particular, we identify the extremizer in an autoconvolution problem posed by B. Green in 2001.