Reproducing kernels of Sobolev spaces on Rd and applications to embedding constants and tractability

The standard Sobolev space W-2(s) (R-d), with arbitrary positive integers s and d for which s > d/2, has the reproducing kernel K-d,K-s(x, t) = integral(Rd) Pi(d)(j=1) cos(2 pi(x(j) - t(j))u(j))/1 + Sigma(0<vertical bar alpha vertical bar 1 <= s) Pi(d)(j=1)(2 pi u(j))(2 alpha j) du for all x, t is an element of R-d, where x(j), t(j), u(j), alpha(j) are components of d-variate x, t, u, alpha, and vertical bar alpha vertical bar(1) = Sigma(d)(j=1) alpha(j) with non-negative integers alpha(j). We obtain a more explicit form for the reproducing kernel K-1,K- s and find a closed form for the kernel K-d,K- infinity. Knowing the form of K-d,K- s, we present applications on the best embedding constants between the Sobolev space W-2(s) (R-d) and L-infinity(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.