Universal approximation results for neural networks with non-polynomial activation function over non-compact domains

More precisely, by assuming that the activation function is non-polynomial, we derive universal approximation results for neural networks within function spaces over non-compact subsets of a Euclidean space, e.g., weighted spaces, L Furthermore, we provide some dimension-independent rates for approximating a function with sufficiently regular and integrable Fourier transform by neural networks with non-polynomial activation function. Inspired by the functionality of human brains, (artificial) neural networks have been discovered in the seminal work of McCulloch and Pitts (see [32]). Fundamentally, a neural network consists of nodes arranged in hierarchical layers, where the connections between adjacent layers transmit the data through the network and the nodes transform this information. In mathematical terms, a neural network can therefore be described as a concatenation of affine and non-affine functions. Nowadays, neural networks are successfully applied in the fields of image classification (see e.g.