Approximation of Eigenfunctions in Kernel-based Spaces

TL;DR

This paper investigates the properties of eigenfunctions in kernel-based spaces, demonstrating their optimality, and introduces a numerical method using greedy point selection to approximate eigensystems effectively.

Contribution

It establishes the optimality of eigenspaces in kernel Hilbert spaces and proposes a practical greedy algorithm for their numerical approximation.

Findings

Eigenfunctions have optimality properties among all subspaces of the Hilbert space.

Errors in approximation are closely related to eigenvalue decay.

The greedy point selection method accurately approximates eigensystems.

Abstract

Kernel-based methods in Numerical Analysis have the advantage of yielding optimal recovery processes in the "native" Hilbert space $\calh$ in which they are reproducing. Continuous kernels on compact domains have an expansion into eigenfunctions that are both $L_2$-orthonormal and orthogonal in $\calh$ (Mercer expansion). This paper examines the corresponding eigenspaces and proves that they have optimality properties among all other subspaces of $\calh$. These results have strong connections to $n$-widths in Approximation Theory, and they establish that errors of optimal approximations are closely related to the decay of the eigenvalues. Though the eigenspaces and eigenvalues are not readily available, they can be well approximated using the standard $n$-dimensional subspaces spanned by translates of the kernel with respect to $n$ nodes or centers. We give error bounds for the numerical approximation of the eigensystem via such subspaces. A series of examples shows that our numerical technique via a greedy point selection strategy allows to calculate the eigensystems with good accuracy.