Before revision:  We prove some general results which imply that an artificial neural network with a single hidden layer and sigmoid activation represents no function in L^{p(} ^2 \times 0, 1 ^{n neural network with a single hidden layer and logistic activation is a universal approximator of L^{2}(   \times [0, 1] ^{n^+ \times } [0, 1] ^{n .
After revision:  We prove some general results which imply that an artificial neural network with a single hidden layer and sigmoid activation represents no function in L^{p(} ^2 \times 0, 1 We prove that a shallow neural network with a single hidden layer and logistic activation is a universal approximator of L^{2}(  \times [0, 1] ^{n^+ \times } [0, 1] ^{n .