Reproducing Kernel Hilbert Space vs. Frame Estimates

TL;DR

This paper explores conditions under which a frame of vectors in a Hilbert space induces a reproducing kernel Hilbert space structure, providing explicit formulas and applications to Gaussian process-related spaces.

Contribution

It identifies specific conditions on frame vectors as functions that ensure the Hilbert space becomes a reproducing kernel Hilbert space, with explicit kernel formulas and isomorphism descriptions.

Findings

Derived explicit reproducing kernels for specific frames

Established conditions linking frames to RKHS structures

Applied results to Gaussian process Hilbert spaces

Abstract

We consider conditions on a given system $\mathcal{F}$ of vectors in Hilbert space $\mathcal{H}$, forming a frame, which turn $\mathcal{H}$ into a reproducing kernel Hilbert space. It is assumed that the vectors in $\mathcal{F}$ are functions on some set $\Omega$. We then identify conditions on these functions which automatically give $\mathcal{H}$ the structure of a reproducing kernel Hilbert space of functions on $\Omega$. We further give an explicit formula for the kernel, and for the corresponding isometric isomorphism. Applications are given to Hilbert spaces associated to families of Gaussian processes.