Lp approximation capability of RBF neural networks

Lp approximation capability of radial basis function (RBF) neural networks is investigated. If g: R+1 → R1 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ g(\parallel x\parallel _{R^n } ) $$\end{document} ∈ Llocp(Rn) with 1 ≤ p < ∞, then the RBF neural networks with g as the activation function can approximate any given function in Lp(K) with any accuracy for any compact set K in Rn, if and only if g(x) is not an even polynomial.