Universalities of reproducing kernels revisited

Kernel methods have been widely applied to machine learning and other questions of approximating an unknown function from its finite sample data. To ensure arbitrary accuracy of such an approximation, various denseness conditions are imposed on the selected kernel. This note contributes to the study of universal, characteristic, and C-0-universal kernels. We first give a simple and direct description of the difference and relation among these three kinds of universalities of kernels. We then focus on translation-invariant and Hilbert-Schmidt kernels formed by polynomials. A simple and shorter proof of the known characterization of characteristic translation-invariant kernels will be presented. The main purpose of the note is to give a delicate discussion on the universalities of Hilbert-Schmidt kernels formed by weighted polynomials.