论文标题
多面体在理性参数性超曲面上的傅立叶变换的身份定理
An identity theorem for the Fourier transform of polytopes on rationally parameterisable hypersurfaces
论文作者
论文摘要
如果$ \ Mathbb {r}^n $中的点$ \ MATHCAL {S} $，则称为合理参数的Hypersurface，如果$ \ Mathcal {s} = \boldsymbolσ（\boldsymbolσ（\ \boldsymbolσ）（\ \ mathbf {t}） \ mathbb {r}^{n-1} \ rightarrow \ mathbb {r}^n $是一个vector函数，具有域$ d $，作为组件有理功能。 $ \ mathbb {r}^n $中的广义$ n $ dimensional polytope是有限数量的convex $ n $ dimensional polytopes的结合。 $ \ mathbb {r}^n $在$ f _ {\ Mathcal {p}}（\ Mathbf {s}）= \ int _ {\ int _ {\ Mathcal {p} {\ Mathcal {p {p {p}}中定义了这种广义polytope $ \ mathcal {p} $的傅立叶变换。 e^{ - i \ mathbf {s} \ cdot \ mathbf {x}}} \，\ mathbf {dx} $。我们证明$ f _ {\ Mathcal {p} _1}（\boldsymbolσ（\ MathBf {t}））= f _ {\ Mathcal {p} _2} _2} _2}（\ boldsymbol成立） $ \ MATHCAL {p} _1 = \ MATHCAL {P} _2 $如果$ O $是满足某些定义明确的条件的$ D $的打开子集。此外，我们表明该定理可以应用于不包含线路的二次超曲面，而是至少两个点，即尤其是在球体上。
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.
