Perturbed Fourier uniqueness and interpolation results in higher dimensions

TL;DR

This paper establishes new Fourier interpolation and uniqueness results in higher dimensions, showing that functions and their Fourier transforms vanishing on specific perturbed spheres must be zero, with applications to Heisenberg uniqueness.

Contribution

It extends Fourier uniqueness and interpolation results to perturbed surfaces in all dimensions, generalizing previous work and connecting to Heisenberg uniqueness problems.

Findings

Schwartz functions vanishing on perturbed spheres are zero

Results apply to both radial and non-radial cases

Connections made to Heisenberg uniqueness for the hyperbola

Abstract

We obtain new Fourier interpolation and -uniqueness results in all dimensions, extending methods and results by the first author and M. Sousa, and by the second author. We show that the only Schwartz function which, together with its Fourier transform, vanishes on surfaces close to the origin-centered spheres whose radius are square roots of integers, is the zero function. In the radial case, these surfaces are spheres with perturbed radii, while in the non-radial case, they can be graphs of continuous functions over the sphere. As an application, we translate our perturbed Fourier uniqueness results to perturbed Heisenberg uniqueness for the hyperbola, using the interrelation between these fields introduced and studied by Bakan, Hedenmalm, Montes-Rodriguez, Radchenko and Viazovska.