  Artificial neural networks (ANNs) have become a very powerful tool in the
approximation of high-dimensional functions. Especially, deep ANNs, consisting
of a large number of hidden layers, have been very successfully used in a
series of practical relevant computational problems involving high-dimensional
input data ranging from classification tasks in supervised learning to optimal
decision problems in reinforcement learning. There are also a number of
mathematical results in the scientific literature which study the approximation
capacities of ANNs in the context of high-dimensional target functions. In
particular, there are a series of mathematical results in the scientific
literature which show that sufficiently deep ANNs have the capacity to overcome
the curse of dimensionality in the approximation of certain target function
classes in the sense that the number of parameters of the approximating ANNs
grows at most polynomially in the dimension $d \in \mathbb{N}$ of the target
functions under considerations. In the proofs of several of such
high-dimensional approximation results it is crucial that the involved ANNs are
sufficiently deep and consist a sufficiently large number of hidden layers
which grows in the dimension of the considered target functions. It is the
topic of this work to look a bit more detailed to the deepness of the involved
ANNs in the approximation of high-dimensional target functions. In particular,
the main result of this work proves that there exists a concretely specified
sequence of functions which can be approximated without the curse of
dimensionality by sufficiently deep ANNs but which cannot be approximated
without the curse of dimensionality if the involved ANNs are shallow or not
deep enough.
