  We prove that the $E_8$ root lattice and the Leech lattice are universally
optimal among point configurations in Euclidean spaces of dimensions $8$ and
$24$, respectively. In other words, they minimize energy for every potential
function that is a completely monotonic function of squared distance (for
example, inverse power laws or Gaussians), which is a strong form of robustness
not previously known for any configuration in more than one dimension. This
theorem implies their recently shown optimality as sphere packings, and broadly
generalizes it to allow for long-range interactions.
  The proof uses sharp linear programming bounds for energy. To construct the
optimal auxiliary functions used to attain these bounds, we prove a new
interpolation theorem, which is of independent interest. It reconstructs a
radial Schwartz function $f$ from the values and radial derivatives of $f$ and
its 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 this
theorem, we construct an interpolation basis using integral transforms of
quasimodular forms, generalizing Viazovska's work on sphere packing and placing
it in the context of a more conceptual theory.
