  In this paper we are concerned with the approximation of functions by single
hidden layer neural networks with ReLU activation functions on the unit circle.
In particular, we are interested in the case when the number of data-points
exceeds the number of nodes. We first study the convergence to equilibrium of
the stochastic gradient flow associated with the cost function with a quadratic
penalization. Specifically, we prove a Poincar\'e inequality for a penalized
version of the cost function with explicit constants that are independent of
the data and of the number of nodes. As our penalization biases the weights to
be bounded, this leads us to study how well a network with bounded weights can
approximate a given function of bounded variation (BV).
  Our main contribution concerning approximation of BV functions, is a result
which we call the localization theorem. Specifically, it states that the
expected error of the constrained problem, where the length of the weights are
less than $R$, is of order $R^{-1/9}$ with respect to the unconstrained problem
(the global optimum). The proof is novel in this topic and is inspired by
techniques from regularity theory of elliptic partial differential equations.
Finally we quantify the expected value of the global optimum by proving a
quantitative version of the universal approximation theorem.
