A Note on Compact Embeddings of Reproducing Kernel Hilbert Spaces in L2 and Infinite-Variate Function Approximation

This note consists of two largely independent parts. In the first part we give conditions on the kernel k : Omega x Omega -> R of a reproducing kernel Hilbert space H continuously embedded via the identity mapping into L-2(Omega, mu), which are equivalent to the fact that H is even compactly embedded into L-2(Omega, mu). In the second part we consider a scenario from infinite-variate L-2-approximation. Suppose that the embedding of a reproducing kernel Hilbert space of univariate functions with reproducing kernel 1 + k into L-2(Omega, mu) is compact. We provide a simple criterion for checking compactness of the embedding of a reproducing kernel Hilbert space with the kernel given by Sigma(u is an element of U) gamma(u) circle times(j is an element of u)k where U = {u subset of N : |u| < infinity}, and gamma = (gamma(u))(u is an element of U) is a family of non-negative numbers, into an appropriate L-2 space.