Wiener's tauberian theorems for the Fourier–Bessel transformation and uniform approximation by RBF networks of Delsarte translates

We prove analogues of the Wiener and Wiener–Pitt tauberian theorems for the Delsarte translation and the Fourier–Bessel transformation. Through similar functional-analytic techniques, 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 a certain space of continuous functions endowed with the uniform norm.