  Recently, deep Convolutional Neural Networks (CNNs) have proven to be
successful when employed in areas such as reduced order modeling of
parametrized PDEs. Despite their accuracy and efficiency, the approaches
available in the literature still lack a rigorous justification on their
mathematical foundations. Motivated by this fact, in this paper we derive
rigorous error bounds for the approximation of nonlinear operators by means of
CNN models. More precisely, we address the case in which an operator maps a
finite dimensional input $\boldsymbol{\mu}\in\mathbb{R}^{p}$ onto a functional
output $u_{\boldsymbol{\mu}}:[0,1]^{d}\to\mathbb{R}$, and a neural network
model is used to approximate a discretized version of the input-to-output map.
The resulting error estimates provide a clear interpretation of the
hyperparameters defining the neural network architecture. All the proofs are
constructive, and they ultimately reveal a deep connection between CNNs and the
Fourier transform. Finally, we complement the derived error bounds by numerical
experiments that illustrate their application.
