An identity theorem for the Fourier transform of polytopes on rationally parameterisable hypersurfaces

Abstract

A set $\mathcal{S}$ of points in $\mathbb{R}^n$ is called a rationally parameterisable hypersurface if $\mathcal{S}=\{\boldsymbol{\sigma}(\mathbf{t}): \mathbf{t} \in D\}$, where $\boldsymbol{\sigma}: \mathbb{R}^{n-1} \rightarrow \mathbb{R}^n$ is a vector function with domain $D$ and rational functions as components. A generalized $n$-dimensional polytope in $\mathbb{R}^n$ is a union of a finite number of convex $n$-dimensional polytopes in $\mathbb{R}^n$. The Fourier transform of such a generalized polytope $\mathcal{P}$ in $\mathbb{R}^n$ is defined by $F_{\mathcal{P}}(\mathbf{s})=\int_{\mathcal{P}} e^{-i\mathbf{s}\cdot\mathbf{x}} \,\mathbf{dx}$. We prove that $F_{\mathcal{P}_1}(\boldsymbol{\sigma}(\mathbf{t})) = F_{\mathcal{P}_2}(\boldsymbol{\sigma}(\mathbf{t}))\ \forall \mathbf{t} \in O$ implies $\mathcal{P}_1=\mathcal{P}_2$ if $O$ is an open subset of $D$ satisfying some well-defined conditions. Moreover we show that this theorem can be applied to quadric hypersurfaces that do not contain a line, but at least two points, i.e., in particular to spheres.