Almost positive kernels on compact Riemannian manifolds

Abstract We show how to build a kernel $$ K_X(x,y)=\sum _{m=0}^Xh(\lambda _m/{\lambda _X})\varphi _m(x)\overline{\varphi _m(y)} K X ( x , y ) = ∑ m 0 h λ / φ ¯ on compact Riemannian manifold $${{\,\mathrm{\mathcal {M}}\,}}$$ M , which is positive up negligible error and such that $$K_X(x,x)\approx X$$ ≈ . Here $$0=\lambda _0^2\le \lambda _1^2\le \cdots 2 ≤ 1 ⋯ are the eigenvalues of Laplace–Beltrami operator listed with repetitions, $$\varphi _0,\,\varphi _1,\ldots … an associated system eigenfunctions, forming orthonormal basis $$L^2({{\,\mathrm{\mathcal {M}}\,}})$$ L The function h smooth certain minimal degree, even, compactly supported in $$[-1,1]$$ [ - ] $$h(0)=1$$ $$K_X(x,y)$$ turns out be approximation identity.