  In this article, we prove approximation theorems in classes of deep and
shallow neural networks with analytic activation functions by elementary
arguments. We prove for both real and complex networks with non-polynomial
activation that the closure of the class of neural networks coincides with the
closure of the space of polynomials. The closure can further be characterized
by the Stone-Weierstrass theorem (in the real case) and Mergelyan's theorem (in
the complex case). In the real case, we further prove approximation results for
networks with higher-dimensional harmonic activation and orthogonally projected
linear maps. We further show that fully connected and residual networks of
large depth with polynomial activation functions can approximate any polynomial
under certain width requirements. All proofs are entirely elementary.
