  Bernstein inequalities and inverse theorems are a recent development in the
theory of radial basis function(RBF) approximation. The purpose of this paper
is to extend what is known by deriving $L^p$ Bernstein inequalities for RBF
networks on $\mathbb{R}^d$. These inequalities involve bounding a
Bessel-potential norm of an RBF network by its corresponding $L^p$ norm in
terms of the separation radius associated with the network. The Bernstein
inequalities will then be used to prove the corresponding inverse theorems.
