  Graph convolutional neural network (GCNN) operates on graph domain and it has
achieved a superior performance to accomplish a wide range of tasks. In this
paper, we introduce a Barron space of functions on a compact domain of graph
signals. We prove that the proposed Barron space is a reproducing kernel Banach
space, it can be decomposed into the union of a family of reproducing kernel
Hilbert spaces with neuron kernels, and it could be dense in the space of
continuous functions on the domain. Approximation property is one of the main
principles to design neural networks. In this paper, we show that outputs of
GCNNs are contained in the Barron space and functions in the Barron space can
be well approximated by outputs of some GCNNs in the integrated square and
uniform measurements. We also estimate the Rademacher complexity of functions
with bounded Barron norm and conclude that functions in the Barron space could
be learnt from their random samples efficiently.
