Approximation by RBF neural networks of Hankel translates with a locally $$p$$-integrable activation function

Abstract Radial basis function neural networks (RBFNNs) of Hankel translates are essentially linear combinations of translations and dilations of a so-called activation function defined on the nonnegative real axis, where, instead of the standard translation, the modified Delsarte translation operator associated with the Hankel integral transformation of order $$\mu > -1/2$$ is considered. In this paper, we focus on activation functions $$\sigma$$ such that $$z^{-\mu-1/2}\sigma(z)$$ is locally $$p$$ -integrable with respect to the measure $$z^{2 \mu+1} dz$$ , for $$1\le p < \infty$$ . It is shown that such networks enjoy the universal approximation property, that is, are locally dense in $$p$$ -mean if, and only if, $$z^{-\mu-1/2}\sigma(z)$$ is not an even polynomial. In this way, a result of Nan, Wu, Long, Ma and Sun (2008) that holds true for RBFNNs of standard translates is extended to RBFNNs of Hankel translates.