The even Radon–Kipriyanov transform (Kγ-transform) is suitable for studying problems with the Bessel singular differential operator $${{B}_{{{{\gamma }_{i}}}}} = \frac{{{{\partial }^{2}}}}{{\partial x_{i}^{2}}} + \frac{{{{\gamma }_{i}}}}{{{{x}_{i}}~}}\frac{\partial }{{\partial {{x}_{i}}}},{{\gamma }_{i}} > 0$$. In this work, the odd Radon–Kipriyanov transform and the complete Radon–Kipriyanov transform are introduced to study more general equations containing odd B-derivatives $$\frac{\partial }{{\partial {{x}_{i}}}}~B_{{{{\gamma }_{i}}}}^{k},~~k = 0, 1, 2,~ \ldots $$ (in particular, gradients of functions). Formulas of the Kγ-transforms of singular differential operators are given. Based on the Bessel transforms introduced by B.M. Levitan and the odd Bessel transform introduced by I.A. Kipriyanov and V.V. Katrakhov, a relationship of the complete Radon–Kipriyanov transform with the Fourier transform and the mixed Fourier–Levitan–Kipriyanov–Katrakhov transform is deduced. An analogue of Helgason’s support theorem and an analog of the Paley–Wiener theorem are given.

中文翻译：

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完整的 Radon-Kipriyanov 变换：一些性质

偶数 Radon-Kipriyanov 变换（Kγ-变换）适用于研究贝塞尔奇异微分算子 $${{B}_{{{{\gamma }_{i}}}}} = \frac{{{ {\partial }^{2}}}}{{\partial x_{i}^{2}}} + \frac{{{{\gamma }_{i}}}}{{{{x}_{ i}}~}}\frac{\partial }{{\partial {{x}_{i}}}},{{\gamma }_{i}} > 0$$。在这项工作中，奇数 Radon-Kipriyanov 变换和完全 Radon-Kipriyanov 变换被引入以研究包含奇数 B 导数的更一般方程 $$\frac{\partial }{{\partial {{x}_{i}} }}~B_{{{{\gamma }_{i}}}}^{k},~~k = 0, 1, 2,~ \ldots $$（特别是函数的梯度）。给出了奇异微分算子的Kγ变换公式。基于 BM Levitan 引入的 Bessel 变换和 IA Kipriyanov 和 VV Katrakhov 引入的奇数 Bessel 变换，推导出了完全Radon-Kipriyanov变换与Fourier变换和混合Fourier-Levitan-Kipriyanov-Katrakhov变换的关系。给出了 Helgason 支持定理的模拟和 Paley-Wiener 定理的模拟。