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.
Categories: math.CA