Spectral representations of interpolation spaces of reproducing kernel Hilbert spaces

TL;DR

This paper extends the spectral analysis of interpolation spaces of reproducing kernel Hilbert spaces to all index values, providing new insights into their structure and applications in statistical learning.

Contribution

It generalizes existing spectral decomposition results from the case r=2 to all r, and explores their embedding properties and applications.

Findings

Spectral decomposition of interpolation spaces for all r

Analysis of embedding properties of these spaces

Application to regularisation error estimation in statistical learning

Abstract

In statistical learning theory, interpolation spaces of the form $[\mathrm{L}^2,H]_{\theta,r}$, where $H$ is a reproducing kernel Hilbert space, are in widespread use. So far, however, they are only well understood for fine index $r=2$. We generalise existing results from $r=2$ to all possible values of $r$. In particular, we present a spectral decomposition of such spaces, analyse their embedding properties, and describe connections to the theory of Banach spaces of functions. We additionally present example applications of our results to regularisation error estimation in statistical learning.