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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.
Let $p(cdot): mathbb R^nto(0,infty)$ be a variable exponent function satisfying the globally log-Holder continuous condition. In this article, the authors first obtain a decomposition for any distribution of the variable weak Hardy space into good an
We prove some $L^p$-Liouville theorems for hypoelliptic second order Partial Differential Operators left translation invariant with respect to a Lie group composition law in $mathbb{R}^n$. Results for both solutions and subsolutions are given.
Wavelet set wavelets were the first examples of wavelets that may not have associated multiresolution analyses. Furthermore, they provided examples of complete orthonormal wavelet systems in $L^2(mathbb{R}^d)$ which only require a single generating w
Adjacent dyadic systems are pivotal in analysis and related fields to study continuous objects via collections of dyadic ones. In our prior work (joint with Jiang, Olson and Wei) we describe precise necessary and sufficient conditions for two dyadic
Let ${mathcal P}_k$ denote the set of all algebraic polynomials of degree at most $k$ with real coefficients. Let ${mathcal P}_{n,k}$ be the set of all algebraic polynomials of degree at most $n+k$ having exactly $n+1$ zeros at $0$. Let $$|f|_A := su