  This article derives an accurate, explicit, and numerically stable
approximation to the kernel quadrature weights in one dimension and on tensor
product grids when the kernel and integration measure are Gaussian. The
approximation is based on use of scaled Gauss-Hermite nodes and truncation of
the Mercer eigendecomposition of the Gaussian kernel. Numerical evidence
indicates that both the kernel quadrature and the approximate weights at these
nodes are positive. An exponential rate of convergence for functions in the
reproducing kernel Hilbert space induced by the Gaussian kernel is proved under
an assumption on growth of the sum of absolute values of the approximate
weights.
