Countable Tensor Products of Hermite Spaces and Spaces of Gaussian Kernels

TL;DR

This paper explores countably infinite tensor products of Hermite and Gaussian RKHSs, revealing their structural relationships and isometries, with implications for multivariate problem tractability.

Contribution

It establishes an explicit isometric isomorphism between tensor products of Gaussian and Hermite spaces, extending finite tensor product results to infinite cases.

Findings

Countably infinite tensor products can be identified with RKHSs on proper subsets of sequence spaces.

Tensor products of Gaussian spaces with square-summable parameters are isometrically isomorphic to Hermite space tensor products.

Regularity results for Hermite spaces of single-variable functions are provided.

Abstract

In recent years finite tensor products of reproducing kernel Hilbert spaces (RKHSs) of Gaussian kernels on the one hand and of Hermite spaces on the other hand have been considered in tractability analysis of multivariate problems. In the present paper we study countably infinite tensor products for both types of spaces. We show that the incomplete tensor product in the sense of von Neumann may be identified with an RKHS whose domain is a proper subset of the sequence space $\mathbb{R}^\mathbb{N}$. Moreover, we show that each tensor product of spaces of Gaussian kernels having square-summable shape parameters is isometrically isomorphic to a tensor product of Hermite spaces; the corresponding isomorphism is given explicitly, respects point evaluations, and is also an $L^2$-isometry. This result directly transfers to the case of finite tensor products. Furthermore, we provide regularity results for Hermite spaces of functions of a single variable.