Fourier interpolation from spheres.

In every dimension d ≥ 2, we give an explicit formula that expresses the values of any Schwartz function on Rd 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 [Publ. Math. Inst. Hautes Études Sci. 129 (2019), pp. 51–81] 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 Poincaré 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 [ Fourier interpolation with zeros of zeta and L-functions , arXiv:2005.02996, 2020] [ABSTRACT FROM AUTHOR]