On the speed of uniform convergence in Mercer's theorem

TL;DR

This paper analyzes the rate at which the series in Mercer's theorem converges uniformly, relating it to eigenvalue decay and differentiability of kernels, with applications to spectral characterizations of integral operators.

Contribution

It provides explicit bounds on the convergence speed of Mercer's series based on eigenvalue decay and kernel smoothness, extending understanding of spectral properties.

Findings

Convergence speed depends on eigenvalue decay rate.

For 2m times differentiable kernels, approximation error scales with eigenvalue sums.

Applications include spectral characterization of integral operators.

Abstract

The classical Mercer's theorem claims that a continuous positive definite kernel $K({\mathbf x}, {\mathbf y})$ on a compact set can be represented as $\sum_{i=1}^\infty \lambda_i\phi_i({\mathbf x})\phi_i({\mathbf y})$ where $\{(\lambda_i,\phi_i)\}$ are eigenvalue-eigenvector pairs of the corresponding integral operator. This infinite representation is known to converge uniformly to the kernel $K$. We estimate the speed of this convergence in terms of the decay rate of eigenvalues and demonstrate that for $2m$ times differentiable kernels the first $N$ terms of the series approximate $K$ as $\mathcal{O}\big((\sum_{i=N+1}^\infty\lambda_i)^{\frac{m}{m+n}}\big)$ or $\mathcal{O}\big((\sum_{i=N+1}^\infty\lambda^2_i)^{\frac{m}{2m+n}}\big)$. Finally, we demonstrate some applications of our results to a spectral charaterization of integral operators with continuous roots and other powers.