Neural network integral representations with the ReLU activation function

We derive a formula for the integral representation of a shallow neural network with the ReLU activation function under the finite $L_1$-norm assumption on the outer weights with respect to Lebesgue measure on the sphere. In the case of univariate target functions, we further provide a closed-form formula for all possible representations. Additionally, in this case, our formula allows one to explicitly solve the least $L_1$-norm neural network representation for a given function.