L(p) approximation capabilities of sum-of-product and sigma-pi-sigma neural networks

This paper studies the L(p) approximation capabilities of sum-of-product (SOPNN) and sigma-pi-sigma (SPSNN) neural networks. It is proved that the set of functions that are generated by the SOPNN with its activation function in $L_{loc};p(mathcal{R})$ is dense in $L;p(mathcal{K})$ for any compact set $mathcal{K}subset mathcal{R};N$, if and only if the activation function is not a polynomial almost everywhere. It is also shown that if the activation function of the SPSNN is in ${L_{loc};infty(mathcal{R})}$, then the functions generated by the SPSNN are dense in $L;p(mathcal{K})$ if and only if the activation function is not a constant (a.e.).