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 system
of continuous functions, ϕ0(x)=1 for all x∈X,
{ℓk}k=0∞ be an nondecreasing sequence of real numbers with
ℓ0=1, ℓk↑∞ as k→∞, ΠL:=span{ϕj:ℓj≤L}, L≥0. We describe conditions to ensure an
equivalence between the Lp 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.Comment: 28 pages, submitted for publicatio