Embeddings of Reproducing Kernel Hilbert Spaces with General Weights

We study embeddings between reproducing kernel Hilbert spaces $H(K)$ of functions of $d\in \mathbb{N}\cup \{\mathrm{\infty }\}$ 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 ($d=\mathrm{\infty }$) 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.