Universal optimality of the $E_8$ and Leech lattices and interpolation formulas

We prove that the $E_8$ root lattice and the Leech lattice are universallyoptimal among point configurations in Euclidean spaces of dimensions $8$ and$24$, respectively. In other words, they minimize energy for every potentialfunction that is a completely monotonic function of squared distance (forexample, inverse power laws or Gaussians), which is a strong form of robustnessnot previously known for any configuration in more than one dimension. Thistheorem implies their recently shown optimality as sphere packings, and broadlygeneralizes it to allow for long-range interactions. The proof uses sharp linear programming bounds for energy. To construct theoptimal auxiliary functions used to attain these bounds, we prove a newinterpolation theorem, which is of independent interest. It reconstructs aradial Schwartz function $f$ from the values and radial derivatives of $f$ andits Fourier transform $\widehat{f}$ at the radii $\sqrt{2n}$ for integers$n\ge1$ in $\mathbb{R}^8$ and $n \ge 2$ in $\mathbb{R}^{24}$. To prove thistheorem, we construct an interpolation basis using integral transforms ofquasimodular forms, generalizing Viazovska's work on sphere packing and placingit in the context of a more conceptual theory.