LpLp Bernstein inequalities and inverse theorems for RBF approximation on RdRd

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 LpLp Bernstein inequalities for RBF networks on RdRd. These inequalities involve bounding a Bessel-potential norm of an RBF network by its corresponding LpLp norm in terms of the separation radius associated with the network. The Bernstein inequalities will then be used to prove the corresponding inverse theorem.