论文标题
从傅立叶变换的模量的重建多型的，小波长
Reconstruction of polytopes from the modulus of the Fourier transform with small wave length
论文作者
论文摘要
令$ \ mathcal {p} $为$ n $ -Dimensional convex polytope，而$ \ Mathcal {s} $是$ \ Mathbb {r}^n $中的HyperSurface。本文研究了重建$ \ MATHCAL {P} $的潜力，或者至少要计算$ \ Mathcal {p} $的重要属性，如果$ \ Mathcal {p} $的傅立叶变换的模量在$ \ Mathcal {s} $ in $ \ MATHCAL {s} $带有波长$ cullume $ cullength $ cullengti e^{ - i \ frac {1}λ\ mathbf {s} \ cdot \ cdot \ mathbf {x}}}} \，\ mathbf {dx} | $ for $ \ mathbf {s} \ in \ mathcal in \ mathcal {s} $，给出了{ $ \ Mathcal {S} $具有一些定义明确的属性。主要工具是用于$ \ MATHCAL {p} $的傅里叶变换的渐近公式，波长$λ$当$λ\ rightarrow 0 $。纳米颗粒的X射线散射理论激发了这项研究，因为在实验中，反射梁波矢量的傅立叶变换的模量大致可测量。
Let $\mathcal{P}$ be an $n$-dimensional convex polytope and $\mathcal{S}$ be a hypersurface in $\mathbb{R}^n$. This paper investigates potentials to reconstruct $\mathcal{P}$ or at least to compute significant properties of $\mathcal{P}$ if the modulus of the Fourier transform of $\mathcal{P}$ on $\mathcal{S}$ with wave length $λ$, i.e., $|\int_{\mathcal{P}} e^{-i\frac{1}λ\mathbf{s}\cdot\mathbf{x}} \,\mathbf{dx}|$ for $\mathbf{s}\in\mathcal{S}$, is given, $λ$ is sufficiently small and $\mathcal{P}$ and $\mathcal{S}$ have some well-defined properties. The main tool is an asymptotic formula for the Fourier transform of $\mathcal{P}$ with wave length $λ$ when $λ\rightarrow 0$. The theory of X-ray scattering of nanoparticles motivates this study since the modulus of the Fourier transform of the reflected beam wave vectors are approximately measurable in experiments.
