  In every dimension $d \geq 2$, we give an explicit formula that expresses the
values of any Schwartz function on $\mathbb{R}^d$ only in terms of its
restrictions, and the restrictions of its Fourier transform, to all
origin-centered spheres whose radius is the square root of an integer. We thus
generalize an interpolation theorem by Radchenko and Viazovska to higher
dimensions. We develop a general tool to translate Fourier uniqueness- and
interpolation results for radial functions in higher dimensions, to
corresponding results for non-radial functions in a fixed dimension. In
dimensions greater or equal to 5, we solve the radial problem using a
construction closely related to classical Poincare series. In the remaining
small dimensions, we combine this technique with a direct generalization of the
Radchenko--Viazovska formula to higher-dimensional radial functions, which we
deduce from general results by Bondarenko, Radchenko and Seip.
