  We study the power of deep neural networks (DNNs) with sigmoid activation
function. Recently, it was shown that DNNs approximate any $d$-dimensional,
smooth function on a compact set with a rate of order $W^{-p/d}$, where $W$ is
the number of nonzero weights in the network and $p$ is the smoothness of the
function. Unfortunately, these rates only hold for a special class of sparsely
connected DNNs. We ask ourselves if we can show the same approximation rate for
a simpler and more general class, i.e., DNNs which are only defined by its
width and depth. In this article we show that DNNs with fixed depth and a width
of order $M^d$ achieve an approximation rate of $M^{-2p}$. As a conclusion we
quantitatively characterize the approximation power of DNNs in terms of the
overall weights $W_0$ in the network and show an approximation rate of
$W_0^{-p/d}$. This more general result finally helps us to understand which
network topology guarantees a special target accuracy.
