Almost positive kernels on compact Riemannian manifolds

Abstract

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