<p>Graph convolutional neural network (GCNN) offers a framework for representing and learning functions of data on networks and irregular domains. In this paper, we introduce the concept of a graph Barron space of functions. We prove that the proposed graph Barron space is a reproducing kernel Banach space, can be decomposed into a union of reproducing kernel Hilbert spaces with explicitly expressed neuron kernels, and is dense in the space of continuous functions under certain technical assumptions. In this paper, we also show that the outputs of shallow GCNNs belong to the graph Barron space and that functions in the graph Barron space can be well approximated by outputs of shallow GCNN in both the integrated square and uniform norms. Moreover, we estimate the Rademacher complexity of functions with bounded Barron norm and conclude that functions in the graph Barron space can be learned efficiently from their noiseless random samples with high probability. Finally, we test the approximation performance of shallow GCNNs to a quadratic function on the data set collected at weather stations in the region of Brest, France.</p>