Almost positive kernels on compact Riemannian manifolds

We show how to build a kernel K_X(x, y) = \Sigma_(m=0)^X h(lambda_(m)/lambda_(X))phi_(m)(x)phi_(m)(y) on a compact Riemannian manifold M, which is positive up to a negligible error and such that K_X(x, x) approximate to X. Here 0 = lambda_(0) <= lambda_(1) <= ... are the eigenvalues of the Laplace-Beltrami operator on M, listed with repetitions, and phi_(0), phi_(1), ... 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.