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 $\mathcal{S}=\{\mathbf{\sigma }(\mathbf{t}):\mathbf{t}\in D\}$, where $\mathbf{\sigma }:{\mathbb{R}}^{n-1}\to {\mathbb{R}}^n$ 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 $F_{{\mathcal{P}}_1}(\mathbf{\sigma }(\mathbf{t}))=F_{{\mathcal{P}}_2}(\mathbf{\sigma }(\mathbf{t}))\forall \mathbf{t}\in O$ 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.

20 pages