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$L^p$ Bernstein Inequalities and Inverse Theorems for RBF Approximation on $\mathbb{R}^d$

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.