电流的X射线和形状的投影
X-rays of currents and projections of forms
摘要：
我们引入并研究了一种新的Radon-like变换，它平均了R^n中仿射（n-k）平面上的投影微分p-形式。然后，我们证明了一个明确的反演公式我们的快速衰减光滑p-形式的空间上的变换。我们的变换不同于Gelfand-Graev-Shapiro中的变换。此外，如果它可以扩展到更大的p-形式空间，我们的反演公式将允许将R^n上任何快速衰减的光滑p-形式合成为k维子空间上p-形式的拉回的（连续）叠加。反过来，这样的综合意味着一个明确的公式（我们推导出），用于重建R^n中的紧支电流（例如，紧定向k维子簇）从它们在k平面上的定向投影。
英文摘要：
We introduce and study a new Radon-like transform that averages projected differential p-forms in R^n over affine (n-k)-planes. We then prove an explicit inversion formula for our transform on the space of rapidly-decaying smooth p-forms. Our transform differs from the one in Gelfand-Graev-Shapiro. Moreover, if it can be extended to a somewhat larger space of p-forms, our inversion formula will allow the synthesis of any rapidly-decaying smooth p-form on R^n as a (continuous) superposition of pullbacks from p-forms on k-dimensional subspaces. In turn, such synthesis implies an explicit formula (which we derive) for reconstructing compactly supported currents in R^n (e.g., compact oriented k-dimensional subvarieties) from their oriented projections onto k-planes.
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