A short note on compact embeddings of reproducing kernel Hilbert spaces in $L^2$ for infinite-variate function approximation

TL;DR

This paper investigates conditions for compact embeddings of reproducing kernel Hilbert spaces into L^2 spaces, with applications to infinite-variate function approximation and criteria for kernel sum structures.

Contribution

It provides new conditions for compact embedding of RKHS into L^2 and a criterion for the compactness of sum-structured kernels in infinite-variate settings.

Findings

Characterization of kernel conditions for compact embedding.

A simple criterion for sum-structured kernel compactness.

Application to infinite-variate L^2-approximation.

Abstract

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 $$\sum_{u \in \mathcal{U}} \gamma_u \bigotimes_{j \in u}k,$$ where $\mathcal{U} = \{u \subset \mathbb{N}: |u| < \infty\},$ and $(\gamma_u)_{u \in \mathcal{U}}$ is a sequence of non-negative numbers, into an appropriate $L^2$ space.