  This article contributes to the current statistical theory of deep neural
networks (DNNs). It was shown that DNNs are able to circumvent the so--called
curse of dimensionality in case that suitable restrictions on the structure of
the regression function hold. In most of those results the tuning parameter is
the sparsity of the network, which describes the number of non-zero weights in
the network. This constraint seemed to be the key factor for the good rate of
convergence results. Recently, the assumption was disproved. In particular, it
was shown that simple fully connected DNNs can achieve the same rate of
convergence. Those fully connected DNNs are based on the unbounded ReLU
activation function. In this article we extend the results to smooth activation
functions, i.e., to the sigmoid activation function. It is shown that
estimators based on fully connected DNNs with sigmoid activation function also
achieve the minimax rates of convergence (up to $\ln n$-factors). In our result
the number of hidden layers is fixed, the number of neurons per layer tends to
infinity for sample size tending to infinity and a bound for the weights in the
network is given.
