Fourier interpolation from spheres

Abstract

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.