Embeddings of Reproducing Kernel Hilbert Spaces with General Weights

TL;DR

This paper investigates embeddings between reproducing kernel Hilbert spaces with kernels formed by weighted tensor products, using a discrete calculus on weights to analyze transformations and applications in numerical tasks.

Contribution

It introduces a novel approach to embedding analysis via weight transformations and develops a discrete calculus on the cone of weights for finite and infinite dimensions.

Findings

Embedding conditions depend on weight transformations.

Discrete calculus on weights aids in understanding kernel embeddings.

Applications include numerical integration and function recovery.

Abstract

We study embeddings between reproducing kernel Hilbert spaces $H(K)$ of functions of $d \in \mathbb{N} \cup \{\infty\}$ variables. The kernels $K$ are superpositions of weighted finite tensor products of a fixed univariate kernel. The basic idea for the embeddings is to compensate a change of the univariate kernel by a suitable transformation of the weights. For the proofs we employ ($d \in \mathbb{N}$) and develop ($d = \infty$) a discrete calculus on the cone of all weights, where completely monotone weights play a particular role. We sketch how to apply the embedding results to computational problems, as, e.g., numerical integration or function recovery.