Approximation capabilities of neural networks on unbounded domains

We prove universal approximation theorems of neural networks in $L^{p}(\mathbb{R} \times [0, 1]^n)$, under the conditions that $p \in [2, \infty)$ and that the activiation function belongs to among others a monotone sigmoid, relu, elu, softplus or leaky relu. Our results partially generalize classical universal approximation theorems on $[0,1]^n.$