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Marcinkiewicz--Zygmund measures on manifolds

Abstract

Let X{\mathbb X} be a compact, connected, Riemannian manifold (without boundary), ρ\rho be the geodesic distance on X{\mathbb X}, μ\mu be a probability measure on X{\mathbb X}, and {ϕk}\{\phi_k\} be an orthonormal system of continuous functions, ϕ0(x)=1\phi_0(x)=1 for all xXx\in{\mathbb X}, {k}k=0\{\ell_k\}_{k=0}^\infty be an nondecreasing sequence of real numbers with 0=1\ell_0=1, k\ell_k\uparrow\infty as kk\to\infty, ΠL:=span{ϕj:jL}\Pi_L:={\mathsf {span}}\{\phi_j : \ell_j\le L\}, L0L\ge 0. We describe conditions to ensure an equivalence between the LpL^p norms of elements of ΠL\Pi_L with their suitably discretized versions. We also give intrinsic criteria to determine if any system of weights and nodes allows such inequalities. The results are stated in a very general form, applicable for example, when the discretization of the integrals is based on weighted averages of the elements of ΠL\Pi_L on geodesic balls rather than point evaluations.Comment: 28 pages, submitted for publicatio

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