The well-known universal approximation theorem states that neural network with one hidden layer can approximate any continuous function on every compact subset of $\mathbb{R}^d$.
My question is whether there is any paper which considers the approximation on the whole $\mathbb{R}^d$ domain?
In my opinion, since the main themes of neural network are image recognition and natural language processing,
it is enough to consider functions on a compact subset.
However along with its great success, its application field is now widely opened to problems based on whole $\mathbb{R}^d$ domain.
Although I can find some papers (Chen and Chen, 1990; Ito, 1992),
they cannot tackle with whole $\mathbb{R}^d$ domain,
because the former introduces "extended real line" $\bar{\mathbb{R}^d}$ instead of $\mathbb{R}^d$,
and the latter considers only continuous function with compact support.
If you know the related paper, could you please tell me?
