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2024 | OriginalPaper | Chapter

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

Author : Marcin Wnuk

Published in: Monte Carlo and Quasi-Monte Carlo Methods

Publisher: Springer International Publishing

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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

$$\begin{aligned} \sum _{u \in \mathcal {U}} \gamma _u \bigotimes _{j \in u}k, \end{aligned}$$

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.

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previous chapter Multilevel MCMC with Level-Dependent Data in a Model Case of Structural Damage Assessment

The mapping S, strictly speaking, needs not to be an embedding due to the potential lack of injectivity. It may happen that two different elements $f,g \in H$ will be mapped into the same equivalence class $[f] \in L^2.$ Results obtained in this paper hold also in this case. However, to keep the nomenclature simple, we will refer to S as to the (identical) embedding.

A simple calculation shows that for any orthonormal basis $(e_j)_{j \in \mathbb {N}}$ of $L^2(\Omega )$ one has $\sum _{j = 1}^{\infty }\Vert T_{L^2,L^2}e_j \Vert _{L^2}^2 = \Vert k \Vert _{L^2(\Omega \times \Omega )}^2$, so $T_{L^2,L^2}$ is even a Hilbert-Schmidt operator, see e.g. Theorem 4.27. from [1].

Note that $\frac{\mu _i}{\nu _i}, i \in \mathbb {N},$ are eigenvalues of $S^*S$ corresponding to eigenvectors forming an orthonormal basis of $\ell ^2(\nu ),$ and thus by the spectral theorem $S^*S$ is compact if and only if $\lim _{i \rightarrow \infty }\frac{\mu _i}{\nu _i} = 0.$ As compactness of S is equivalent to compactness of $S^*S$ the statement follows.

Continouity of $S_{\gamma }$ is equivalent to $H_{\gamma } \subset L^2(\mathcal {X}, \mu ^{\mathbb {N}}),$ so $S_{\gamma }$ is continuous always when it is well-defined.

Note that $\bigcup _n H_n$ is dense in $H_{\gamma }.$

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Title: A Note on Compact Embeddings of Reproducing Kernel Hilbert Spaces in and Infinite-Variate Function Approximation
Author: Marcin Wnuk
Publisher: Springer International Publishing
Book: Monte Carlo and Quasi-Monte Carlo Methods
Print ISBN: 978-3-031-59761-9

Electronic ISBN: 978-3-031-59762-6

Copyright Year: 2024
DOI: https://doi.org/10.1007/978-3-031-59762-6_33

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

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