A Note on Compact Embeddings of Reproducing Kernel Hilbert Spaces in \(L^2\) and Infinite-Variate Function Approximation

This note consists of two largely independent parts. In the first part we give conditions on the kernel \(k: \Omega \times \Omega \rightarrow \mathbb {R}\) 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 \(\begin{aligned} \sum _{u \in \mathcal {U}} \gamma _u \bigotimes _{j \in u}k, \end{aligned}\) where \(\mathcal {U} = \{u \subset \mathbb {N}: |u| < \infty \},\) and \(\gamma = (\gamma _u)_{u \in \mathcal {U}}\) is a family of non-negative numbers, into an appropriate \(L^2\) space.