  In this effort, we derive a formula for the integral representation of a
shallow neural network with the ReLU activation function. We assume that the
outer weighs admit a finite $L_1$-norm with respect to Lebesgue measure on the
sphere. For 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.
