C^∞ compactly supported and positive definite radial kernels

A family of C^∞ compactly supported radial kernels is presented. These positive definite kernels can be generated numerically using convolutions of compactly supported radial functions. An alternative proof that shows the infinitely smooth limit of the well-known Wendland functions is not a suitable C^∞ compactly supported kernel is also presented. Numerical experiments are provided to demonstrate the accuracy of approximations of the proposed kernels, including comparisons with Gaussians and Wendland radial basis functions of finite smoothness.