Marcinkiewicz-Zygmund measures on manifolds

Let X be a compact, connected, Riemannian manifold (without boundary), ρ be the geodesic distance on X, μ be a probability measure on X, and {φk} be an orthonormal (with respect to μ) system of continuous functions, φ0(x) = 1 for all x ∈ X, {`k} ∞ k=0 be an nondecreasing sequence of real numbers with `0 = 1, `k ↑ ∞ as k → ∞, ΠL := span {φj : `j ≤ L}, L ≥ 0. We describe conditions to ensure an equivalence between the L norms of elements of ΠL 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 on geodesic balls rather than point evaluations.