We consider two families of Funk-type transforms that assign to a function on the unit sphere the integrals of that function over spherical sections by planes of fixed dimension. Transforms of the first kind are generated by planes passing through a fixed center outside the sphere. Similar transforms with interior center and with center on the sphere itself we studied in previous publications. Transforms of the second kind, or the parallel slice transforms, correspond to planes that are parallel to a fixed direction. We show that the Funk-type transforms with exterior center express through the parallel slice transforms and the latter are intimately related to the Radon–John d-plane transforms on the Euclidean ball. These results allow us to investigate injectivity of our transforms and obtain inversion formulas for them. We also establish connection between the Funk-type transforms of different dimensions with arbitrary center.

中文翻译：

---

关于两个Funk型变换族

我们考虑两个Funk型变换族，它们将单位球面上的函数的积分通过固定尺寸的平面分配给单位球面上的函数积分。第一类变换是通过穿过球体外部固定中心的平面生成的。我们在先前的出版物中研究了内部中心和球体中心相似的变换。第二种变换或并行切片变换对应于平行于固定方向的平面。我们表明，具有外部中心的Funk型变换通过平行切片变换来表达，而后者与欧几里得球上的Radon-John d平​​面变换密切相关。这些结果使我们能够研究变换的内射性并为其获得反演公式。