Abstract We apply Erdélyi–Kober fractional integrals to the study of Radon type transforms that take functions on the Grassmannian of j-dimensional affine planes in ℝn to functions on a similar manifold of k-dimensional planes by integration over the set of all j-planes that meet a given k-plane at a right angle. We obtain explicit inversion formulas for these transforms in the class of radial functions under minimal assumptions for all admissible dimensions. The general (not necessarily radial) case, but for j + k = n − 1, n odd, was studied by S. Helgason [8] and F. Gonzalez [4, 5] on smooth compactly supported functions.

中文翻译：

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Erdélyi-Kober 分数积分和相互正交仿射平面的氡变换

摘要 我们将 Erdélyi-Kober 分数积分应用于 Radon 型变换的研究，该变换将 ℝn 中 j 维仿射平面的格拉斯曼函数带到 k 维平面的相似流形上，通过对所有 j 平面的集合进行积分以直角与给定的 k 平面相交。我们在所有可接受维度的最小假设下获得径向函数类中这些变换的显式反演公式。S. Helgason [8] 和 F. Gonzalez [4, 5] 研究了一般（不一定是径向的）情况，但对于 j + k = n − 1，n 奇数，研究了平滑紧支持函数。