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This work suggests using sampling theory to analyze the function spacerepresented by neural networks. First, it shows, under the assumption of afinite input domain, which is the common case in training neural networks, thatthe function space generated by multi-layer networks with non-expansiveactivation functions is smooth. This extends over previous works that showresults for the case of infinite width ReLU networks. Then, under theassumption that the input is band-limited, we provide novel error bounds forunivariate neural networks. We analyze both deterministic uniform and randomsampling showing the advantage of the former.

A function space analysis of finite neural networks with insights from sampling theory