Title: Multiplicative convolution with symmetries in Euclidean space and on the sphere Show Chinese title

Title: 欧几里得空间和球面上具有对称性的乘法卷积

Abstract: Multiplicative convolution $\mu \ast \nu$ of two finite signed measures $\mu$ and $\nu$ on $\mathbb{R}^n$ and a related product $\mu \circledast \nu$ on the sphere $S^{n-1}$ are studied. For fixed $\mu$ the injectivity in $\nu$ of both operations is characterised given an arbitrary group of reflections along the coordinate axes. The results for the sphere yield generalised versions of the theorems in Molchanov and Nagel (2021) about convex bodies. Show Chinese abstract

Abstract: 研究了定义在$\mathbb{R}^n$上的两个有限符号测度$\mu$和$\nu$的乘法卷积$\mu \ast \nu$以及与之相关的球面$S^{n-1}$上的乘积$\mu \circledast \nu$。 对于固定的$\mu$，两种运算在$\nu$中的单射性由沿坐标轴的任意反射群来刻画。 关于球的结果推广了 Molchanov 和 Nagel (2021) 关于凸体的定理。