Denseness of radial-basis functions in \(L^ 2(R^ n)\) and its applications in neural networks (Q1917815)

The authors begin with the following result. Suppose that \(g:\mathbb{R}^i\mapsto\mathbb{R}^1\) and \((1+|x|)^{(n-1)/2}g\in L^2(\mathbb{R}^+)\). The set of linear combinations \(\sum^n_{i=1} c_ig(\lambda_i|X-X_i|_{\mathbb{R}^n})\) is dense in \(L^2(\mathbb{R}^n)\), where \(\lambda_i>0\), \(c_i\in \mathbb{R}^1\), \(X_i\in \mathbb{R}^n\) \((i=1,\dots, N)\). They proceed to discuss approximation to functions in \(L^2(\mathbb{R}^n)\) and operators from \(L^2(\mathbb{R}^{n_2})\) to \(L^2(\mathbb{R}^{n_2})\) by radial-basis functions. The results obtained solve the problem of capability of RBF neural networks, a basic problem in neural networks. For the background to this research see \textit{M. J. D. Powell} [Numerical analysis, Proc. 12th Dundee Bienn. Conf., Dundee/UK 1987, Pitman Res. Notes Math. Ser. 170, 223-241 (1988; Zbl 0687.41004)].