Let $K: \boldsymbol{\Omega}\times \boldsymbol{\Omega}$ be a continuous Mercer kernel defined on a compact subset of ${\mathbb R}^n$ and $\mathcal{H}_K$ be the reproducing kernel Hilbert space (RKHS) associated with $K$. Given a finite measure $\nu$ on $\boldsymbol{\Omega}$, we investigate upper and lower bounds on the $\varepsilon$-entropy of the unit ball of $\mathcal{H}_K$ in the space $L_p(\nu)$. This topic is an important direction in the modern statistical theory of kernel-based methods. We prove sharp upper and lower bounds for $p\in [1,+\infty]$. For $p\in [1,2]$, the upper bounds are determined solely by the eigenvalue behaviour of the corresponding integral operator $\phi\to \int_{\boldsymbol{\Omega}} K(\cdot,{\mathbf y})\phi({\mathbf y})d\nu({\mathbf y})$. In constrast, for $p>2$, the bounds additionally depend on the convergence rate of the truncated Mercer series to the kernel $K$ in the $L_p(\nu)$-norm. We discuss a number of consequences of our bounds and show that they are substantially tighter than previous bounds for general kernels. Furthermore, for specific cases, such as zonal kernels and the Gaussian kernel on a box, our bounds are asymptotically tight as $\varepsilon\to +0$.

翻译：设$K: \boldsymbol{\Omega}\times \boldsymbol{\Omega}$为定义在${\mathbb R}^n$紧子集上的连续Mercer核，$\mathcal{H}_K$为与之关联的再生核希尔伯特空间（RKHS）。给定$\boldsymbol{\Omega}$上的有限测度$\nu$，我们研究$\mathcal{H}_K$单位球在$L_p(\nu)$空间中$\varepsilon$-熵的上下界。该课题是现代核方法统计理论的重要研究方向。我们证明了$p\in [1,+\infty]$情形下的精确上下界。对于$p\in [1,2]$，上界完全由相应积分算子$\phi\to \int_{\boldsymbol{\Omega}} K(\cdot,{\mathbf y})\phi({\mathbf y})d\nu({\mathbf y})$的特征值行为决定。相反地，对于$p>2$，其界还依赖于截断Mercer级数在$L_p(\nu)$范数下逼近核函数$K$的收敛速率。我们讨论了所得界的若干推论，并证明其相较于一般核的已有界显著更紧。此外，对于特定情形（如盒区域上的带核与高斯核），当$\varepsilon\to +0$时，我们的界是渐近紧的。