Fourier interpolation from spheres

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 its restrictions, and restrictions Fourier transform, to all origin-centered spheres whose radius is square root integer. We thus generalize interpolation theorem by Radchenko Viazovska [Publ. Math. Inst. Hautes Études Sci. 129 (2019), pp. 51–81] higher dimensions. develop a general tool translate uniqueness results for radial functions dimensions, corresponding non-radial fixed dimension. dimensions greater or equal $5$, solve problem using construction closely related classical Poincaré series. remaining small combine this technique with direct generalization Radchenko–Viazovska higher-dimensional functions, which deduce from Bondarenko, Seip [Fourier zeros zeta L-functions, arXiv:2005.02996, 2020]