Almost positive kernels on compact Riemannian manifolds

We show how to build a kernel $K_X(x,y)=\sum_{m=0}^Xh(λ_m/{λ_X})φ_m(x)\overline{φ_m(y)}$ on a compact Riemannian manifold $M$, which is positive up to a negligible error and such that $K_X(x,x)\approx X$. Here $0=λ_0^2\leλ_1^2\le\ldots$ are the eigenvalues of the Laplace-Beltrami operator on $M$, listed with repetitions, and $φ_0,\,φ_1,\ldots$ 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.

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