论文标题
紧凑的Riemannian歧管上几乎积极的核
Almost positive kernels on compact Riemannian manifolds
论文作者
论文摘要
我们展示了如何构建一个内核\ [k_x（x，y）= \ sum_ {m = 0}^xh（λ_m/{λ_m/{λ_x}）φ_m（x）\ edmepline {φ_m（y）} \]在紧凑的riemannian歧管上的protibold $ k _上，这是$ k $，大约是$ k _ negligrig and。 x $。这里$ 0 =λ_0^2 \leλ_1^2 \ le \ ldots $是$ m $上的laplace-beltrami运算符的特征值，并列出了重复的$ m $，$φ_0，\，\，φ_1，φ_1，\ ldots $ of eigenfunctions of eigenfunctions of or o y orthorm $ l^2（m）功能$ h $平滑至一定的最低度，即使在$ [-1,1] $中，$ h（0）= 1 $和$ k_x（x，y）$ comply compart compact compact of the Identity近似。
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.
