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

TL;DR

This paper derives explicit forms of reproducing kernels for Sobolev spaces on , and applies these to analyze embedding constants and the tractability of high-dimensional integration, showing exponential decay of errors with dimension.

Contribution

It provides explicit formulas for Sobolev space kernels and demonstrates their use in establishing bounds on embedding constants and tractability of integration.

Findings

Embedding constants decay exponentially with dimension

Integration errors decrease exponentially with the number of function evaluations

Achieves strong polynomial tractability for high-dimensional integration

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.