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

Abstract

The standard Sobolev space $W_2^s({\mathbb{R}}^d)$, with arbitrary positive integers $s$ and $d$ for which $s>d/2$, has the reproducing kernel $$K_{d,s}(x,t)={\int }_{{\mathbb{R}}^d}\frac{{\prod }_{j=1}^d\cos \left(2\pi (x_j-t_j)u_j\right)}{1+{\sum }_{0<|\alpha {|}_1\le s}{\prod }_{j=1}^d(2\pi u_j{)}^{2{\alpha }_j}}\mathrm{d}u$$ 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 $W_2^s({\mathbb{R}}^d)$ 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.

26 pages, 1 figure