Lp approximation of Sigma-Pi neural networks

A feedforward Sigma-Pi neural network with a single hidden layer of m neurons is given by mSigma(j=1) cjg (nPi(k=1) xk-thetak(j)/lambdak(j)) where cj, thetak(j), lambdak are elements of R. In this paper, we investigate the approximation of arbitrary functions f: Rn-->R by a Sigma-Pi neural network in the Lp norm. An Lp locally integrable function g(t) can approximate any given function, if and only if g(t) can not be written in the form Sigma(j=1)n Sigma(k=0)m alphajk(ln/t/)(j-1)tk.