Lp approximation of Sigma-Pi neural networks

A feedforward Sigma-Pi neural network with a single hidden layer of m neurons is given by ^mΣ_j=1c_jg(nΠ_k=1x_k-θ_k^j/λ_k^j) where c_j, θ_k^j, λ_k∈R. We investigate the approximation of arbitrary functions f: Rⁿ→R by a Sigma-Pi neural network in the L^p norm. An L^p locally integrable function g(t) can approximate any given function, if and only if g(t) can not be written in the form Σ_j=1ⁿΣ_k=0^mα_jk(ln|t|)^j-1t_k.