The function space defined by deep neural nets

Question

Given a deep net graph and the activation functions on the hidden vertices do we have a description of the function space spanned by it? (even if for some specific architectures and activation functions)

Or conversely : think of being given a function and an error tolerance (say in the sup norm) and a network architecture (with activation functions). Then is it known how to decide if edge weights can be assigned to the net to represent a function within the given error tolerance?

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A few papers which I think addresses related questions are,

• http://arxiv.org/abs/1606.05336v2
• http://arxiv.org/abs/1509.07385
• http://arxiv.org/abs/1512.03965
• http://arxiv.org/abs/1602.04723