  Constructing neural networks for function approximation is a classical and
longstanding topic in approximation theory. In this paper, we aim at
constructing deep neural networks (deep nets for short) with three hidden
layers to approximate smooth and sparse functions. In particular, we prove that
the constructed deep nets can reach the optimal approximation rate in
approximating both smooth and sparse functions with controllable magnitude of
free parameters. Since the saturation that describes the bottleneck of
approximate is an insurmountable problem of constructive neural networks, we
also prove that deepening the neural network with only one more hidden layer
can avoid the saturation. The obtained results underlie advantages of deep nets
and provide theoretical explanations for deep learning.
