Reproducing Kernels of Sobolev Spaces on $\mathbb{R}^d$ and Applications to Embedding Constants and Tractability

Abstract

The standard Sobolev space $W^s_2(\mathbb{R}^d)$, with arbitrary positive integers $s$ and $d$ for which $s>d/2$, has the reproducing kernel $$ K_{d,s}(x,t)=\int_{\mathbb{R}^d}\frac{\prod_{j=1}^d\cos\left(2\pi\,(x_j-t_j)u_j\right)} {1+\sum_{0<|\alpha|_1\le s}\prod_{j=1}^d(2\pi\,u_j)^{2\alpha_j}}\,{\rm d}u $$ for all $x,t\in\mathbb{R}^d$, where $x_j,t_j,u_j,\alpha_j$ are components of $d$-variate $x,t,u,\alpha$, and $|\alpha|_1=\sum_{j=1}^d\alpha_j$ with non-negative integers $\alpha_j$. We obtain a more explicit form for the reproducing kernel $K_{1,s}$ and find a closed form for the kernel $K_{d, \infty}$. Knowing the form of $K_{d,s}$, we present applications on the best embedding constants between the Sobolev space $W^s_2(\mathbb{R}^d)$ and $L_\infty(\mathbb{R}^d)$, and on strong polynomial tractability of integration with an arbitrary probability density. We prove that the best embedding constants are exponentially small in $d$, whereas worst case integration errors of algorithms using $n$ function values are also exponentially small in $d$ and decay at least like $n^{-1/2}$. This yields strong polynomial tractability in the worst case setting for the absolute error criterion.