We provide a strong uniform consistency result with the convergence rate for the kernel density estimation on Riemannian manifolds with Riemann integrable kernels (in the ambient Euclidean space). We also provide a strong uniform consistency result for the kernel density estimation on Riemannian manifolds with Lebesgue integrable kernels. The kernels considered in this paper are different from the kernels in the Vapnik-Chervonenkis class that are frequently considered in statistics society. We illustrate the difference when we apply them to estimate probability density function. We also provide the necessary and sufficient condition for a kernel to be Riemann integrable on a submanifold in the Euclidean space.

翻译：我们对里曼多元体内核密度估计与里曼内核(在周围的欧几里德空间)的内核密度估计的趋同率提供了强烈一致的结果。我们还对里曼多元体内核密度估计与莱贝斯格内核的内核进行了强烈一致的结果。本文中考虑的内核不同于统计社会经常考虑的瓦普尼克-切尔沃南基斯级内核。当我们应用这些内核来估计概率密度功能时,我们说明了两者的差异。我们还为里曼内核在欧克利德空间的子圆形上进行内核作用估计提供了必要和充分的条件。