Matrix analysis
Spherical Marcinkiewicz-Zygmund inequalities and positive quadrature
Mathematics of Computation
Localized Linear Polynomial Operators and Quadrature Formulas on the Sphere
SIAM Journal on Numerical Analysis
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Let X be a compact, connected, Riemannian manifold (without boundary), @r be the geodesic distance on X, @m be a probability measure on X, and {@f"k} be an orthonormal (with respect to @m) system of continuous functions, @f"0(x)=1 for all x@?X, {@?"k}"k"="0^~ be a nondecreasing sequence of real numbers with @?"0=1, @?"k@6~ as k-~, @P"L:=span{@f"j:@?"j@?L}, L=0. We describe conditions to ensure an equivalence between the L^p norms of elements of @P"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 @P"L on geodesic balls rather than point evaluations.