Almost positive kernels on compact Riemannian manifolds

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 {{,mathrm{mathcal {M}},}}, which is positive up to a negligible error and such that K_X(x,x)approx X. Here 0=lambda _0^2le lambda _1^2le cdots are the eigenvalues of the Laplace–Beltrami operator on {{,mathrm{mathcal {M}},}}, listed with repetitions, and varphi _0,,varphi _1,ldots an associated system of eigenfunctions, forming an orthonormal basis of L^2({{,mathrm{mathcal {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.