Almost positive kernels on compact Riemannian manifolds

TL;DR

This paper constructs a kernel on compact Riemannian manifolds that is nearly positive, approximates the identity, and is based on eigenfunctions of the Laplace-Beltrami operator, useful for analysis and approximation tasks.

Contribution

It introduces a method to build almost positive kernels on compact Riemannian manifolds using eigenfunctions, with controlled approximation properties and negligible negativity.

Findings

Kernel approximates the identity operator

Kernel is positive up to negligible error

Kernel value at diagonal scales with X

Abstract

We show how to build a kernel \[ K_X(x,y)=\sum_{m=0}^Xh(\lambda_m/{\lambda_X})\varphi_m(x)\overline{\varphi_m(y)} \] on a compact Riemannian manifold $M$, which is positive up to a negligible error and such that $K_X(x,x)\approx X$. Here $0=\lambda_0^2\le\lambda_1^2\le\ldots$ are the eigenvalues of the Laplace-Beltrami operator on $M$, listed with repetitions, and $\varphi_0,\,\varphi_1,\ldots$ an associated system of eigenfunctions, forming an orthonormal basis of $L^2(M)$. The function $h$ is smooth up to a certain minimal degree, even, compactly supported in $[-1,1]$ with $h(0)=1$, and $K_X(x,y)$ turns out to be an approximation to the identity.