有理参数化超曲面上多面体的傅里叶变换的一个恒等定理
An identity theorem for the Fourier transform of polytopes on rationally parameterisable hypersurfaces
摘要：
$\mathbb{R}^n$中的点集$\mathcal{S}$称为有理可参数化超曲面，如果$\mathcal{S}=\{\boldsymbolσ(\mathbf{t}): \mathbf{t} \in D\}$，其中$\boldsymbolσ: \mathbb{R}^{n-1} \rightarrow \mathbb{R}^n$是以定义域$D$和有理函数为分支的向量函数。$\mathbb{R}^n$中的广义$n$维多面体是$\mathbb{R}^n$中有限个凸$n$维多面体的并集.在$\mathbb{R}^n$中的这种广义多面体$\mathcal{P}$的傅里叶变换由$F_{\mathcal{P}}(\mathbf{s})=\int_{\mathcal{P}} e^{-i\mathbf{s}\cdot\mathbf{x}} \,\mathbf{dx}$定义。证明了当$O$是满足一定条件的$D$的开子集时，$F_{\mathcal{P}_1}(\boldsymbolσ(\mathbf{t})) = F_{\mathcal{P}_2}(\boldsymbolσ(\mathbf{t}))\ \forall \mathbf{t} \in O$蕴涵$\mathcal{P}_1=\mathcal{P}_2$.此外，我们证明了这个定理可以应用于不包含直线，但至少包含两个点的二次超曲面，即，特别是球体。
英文摘要：
A set $\mathcal{S}$ of points in $\mathbb{R}^n$ is called a rationally parameterisable hypersurface if $\mathcal{S}=\{\boldsymbolσ(\mathbf{t}): \mathbf{t} \in D\}$, where $\boldsymbolσ: \mathbb{R}^{n-1} \rightarrow \mathbb{R}^n$ 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 $F_{\mathcal{P}_1}(\boldsymbolσ(\mathbf{t})) = F_{\mathcal{P}_2}(\boldsymbolσ(\mathbf{t}))\ \forall \mathbf{t} \in O$ 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.
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