Approximation in weighted p-mean by RBF networks of Delsarte translates

We give conditions on the kernel function (or activation function) for the family of radial basis function (RBF) neural networks obtained upon replacing the usual translation by the Delsarte one, with not necessarily the same smoothing factor in all kernel nodes, to have the universal approximation property in suitable weighted L p -spaces ( 1 ⩽ p < ∞ ) . A complete characterization of such kernels for p = 1 and p = 2 is provided.