Fourier uniqueness in even dimensions

Abstract

In recent work, methods from the theory of modular forms were used to obtain Fourier uniqueness results in several key dimensions ([Formula: see text]), in which a function could be uniquely reconstructed from the values of it and its Fourier transform on a discrete set, with the striking application of resolving the sphere packing problem in dimensions [Formula: see text] and [Formula: see text] In this short note, we present an alternative approach to such results, viable in even dimensions, based instead on the uniqueness theory for the Klein-Gordon equation. Since the existing method for the Klein-Gordon uniqueness theory is based on the study of iterations of Gauss-type maps, this suggests a connection between the latter and methods involving modular forms. The derivation of Fourier uniqueness from the Klein-Gordon theory supplies conditions on the given test function for Fourier interpolation, which are hoped to be optimal or close to optimal.