# Universal approximation theorem for whole R^d

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?

• I'd recommend looking at classical polynomial approximation theorems. In particular you need to decide on a metric to measure error on the real line (so it's not always infinite!). See eg Hermite polynomials – seanv507 Dec 3 '18 at 9:58