Marcinkiewicz--Zygmund measures on manifolds

1 Introduction

• Classical MZ inequalities compare integrals and weighted samples of trigonometric polynomials, with applications in approximation theory, number theory, and signal processing.The cited survey reviews classical analogues and applications.Source: §1 · p. 1
• For approximation by an operator with kernel ΦL, discretizing its integral using sampled function values yields a natural discrete operator; for Fourier projection, this discretization is called hyperinterpolation.When p=1, inequality (1.2) gives the discrete and continuous operators norms of the same order in L.Source: §1 · p. 2
• For random sampling nodes, the p=∞ inequality can turn probability bounds on sums involving |wy||ΦL(x,y)| into bounds on the discrete operator norm.An example is cited in.Source: §1 · p. 2
• For p=2, the Gram-matrix eigenvalues are bounded below by c1 and above by c2 when the weights satisfy (1.2), with a ratio c2/c1 closer to 1 indicating better conditioning.This links the discretized norm inequality directly to the conditioning of least-squares approximation.Source: §1 · p. 2
• With p=1 and c1,c2 sufficiently close to 1, the inequalities imply existence of positive quadrature formulas exact on ΠcL for some constant c..Source: §1 · p. 2
• The paper establishes conditions for MZ inequalities across 1 ≤ p ≤ ∞ with constants 1−η and 1+η, and gives intrinsic criteria for systems of nodes and weights.The theory includes weighted averages over geodesic balls, not only point evaluations.Source: §1 · p. 3

2 Riemannian manifolds

• A differentiable manifold is described through coordinate charts whose images cover the space and whose transitions are differentiable; the paper assumes the space is Hausdorff with a countable basis.The authors suggest picturing a manifold as a surface, while noting that the definition does not require an ambient Euclidean space.Source: §2 · pp. 3–4
• Tangent vectors are defined by their action on differentiable functions, and the vectors at each point form the tangent space and together constitute the tangent bundle.Source: §2 · p. 4
• A Riemannian metric assigns smoothly varying scalar products to tangent spaces, defining curve lengths and a volume element.A geodesic is a curve whose length is minimal among piecewise differentiable curves with the same endpoints.Source: §2 · p. 4
• The gradient and divergence are introduced as differential operators, and the Laplace–Beltrami operator is elliptic, with a discrete spectrum and orthonormal eigenfunctions.Source: §2 · pp. 4–5

3 Assumptions

• The function spaces ΠL are spans of orthonormal-system elements with ℓj ≤ L, whose members are called diffusion polynomials of degree at most L.The paper also defines geodesic balls B(x,r) and their complements.Source: §3 · p. 5
• The heat kernel and its gradient are assumed to satisfy exponential off-diagonal decay bounds for t in (0,1], with constants κ2 and κ3.The gradient bound is taken with respect to y.Source: §3 · p. 5
• Kordyukov established that the paper’s assumptions hold for heat kernels associated with eigenfunctions of certain second-order elliptic operators, including the Laplace–Beltrami operator under bounded geometry.The cited discussion also notes estimates for other operators.Source: §3 · p. 5
• The assumptions include a Bessel-system condition for every vector field applied to the orthonormal functions; orthonormal systems themselves satisfy the stated Bessel-system definition.For Laplace–Beltrami eigenfunctions, the paper gives vector-field derivatives as an example and identifies N(f) with the L2 norm of div(fF).Source: §3 · pp. 5–6
• The stated assumptions imply a measure homogeneity bound: μ(B(x,R)) ≤ c(R/r)^α μ(B(x,r)) for 0 < r ≤ 1 and R > 0.Source: §3 · p. 6

4 MZ measures

• Writing weighted sums as integrals against a signed measure ν avoids separately carrying the node set and weights through the theory.The notation also accommodates weighted averages over balls and weighted measures dν = wdμ.Source: §4 · pp. 6–7
• For signed measures, total variation captures the absolute masses; for point weights, |ν|(X)=Σy∈C |wy|, and for dν=wdμ, d|ν|=|w|dμ.The support of ν consists of points whose every geodesic ball has positive total variation.Source: §4 · p. 7
• Nikolskii inequalities provide bounds between Lp and Lr norms of diffusion polynomials, while weighted measures with integrable densities yield corresponding inequalities with constants that may depend polynomially on L.The section notes that finite-dimensionality alone gives inequalities whose constants may depend on L.Source: §4 · pp. 7–8
• An MZ measure is one for which the constants in the MZ inequalities are independent of L.Source: §4 · p. 8

• Theorem 5.2 constructs a finite subset of supp(ν) and a partition whose cells lie in 81d-balls and have positive lower bounds on |ν|-mass.The construction assumes δ(supp(ν))<d≤1/81.Source: §5 · p. 9
• Theorem 5.3 gives finite-p and supremum-norm MZ inequalities for signed measures under frequency limits L≤c2η(pd)−1 and L≤c3ηd−1, respectively.The supremum-norm case settles p=∞; the subsequent discussion focuses on 1≤p<∞.Source: §5 · p. 9
• 1/L-regularity is sufficient for the upper Lp norm inequality, and any such inequality implies 1/L-regularity with its regularity constant controlled by the inequality constant.These assertions apply to P∈ΠL and 1≤p<∞.Source: §5 · p. 11
• c4/L-dominance suffices for the reverse Lp norm inequality; conversely, a lower norm bound forces dominance at a scale determined by its constant and the regularity norm.The converse specifies d as c5(S)(max(1, |||ν|||R,1/LA1)^(1/(S−α)))L−1.Source: §5 · p. 11
• If δ(supp(ν))<d<c1 and 0<L<c2d−1, Theorem 5.5 provides a nonnegative weight function on supp(ν) satisfying the stated quadrature condition.The authors note that the condition depends only on supp(ν).Source: §5 · p. 11
• Under the product assumption, mesh-based estimates cover L≤cη/(pδ(C)) in parts (a) and (b), and L≤cη/δ(C) in part (c).The theorem’s constants may depend on κ and A.Source: §5 · pp. 9, 12
• Under the product assumption, a positive quadrature measure of order 2A*L yields the theorem’s norm inequality with constants independent of τ and L.The constants may depend on p.Source: §5 · p. 12

6 Preparatory results

• Section 6 assembles these preparatory tools for the Section 5 proofs, including interpolation, localized kernel properties, and positive-functional extension.The extension result is used in proving existence of positive quadrature formulas.Source: §6 · p. 12

6.1 Riesz–Thorin interpolation theorem

• Section 6.1 proves an interpolation theorem that combines endpoint operator bounds to obtain bounds at intermediate exponents.The theorem relates 1/p and 1/r to the endpoint exponents through the interpolation parameter t.Source: §6.1 · p. 13

6.2 Localized polynomial operators

• Section 6.2 gives localized polynomial-operator estimates under geometric conditions on the node set, including bounds involving mesh size and polynomial scale.The estimates in (6.18) and (6.19) control local suprema or gradients using the L1 norm.Source: §6.2 · pp. 15–16

6.3 Krein’s extension theorem

• Section 6.3 proves that a functional positive on a subspace can be extended to the full space while preserving positivity and a norm bound.The extension theorem assumes a bounded set of dual functionals and a distinguished vector satisfying the stated condition.Source: §6.3 · pp. 16–17

7 Proofs of the results in Section 5.

• The proof of Theorem 5.2 applies a partition-construction lemma successively to balance measure mass, control the signed measure, and ensure each final cell contains a selected node.The resulting cells have measure comparable to d^α and retain a lower bound on their |ν|-mass.Source: §7 · pp. 19–20
• For 1/L-regular ν, the proof transfers Lp bounds between ν and μ for every 1 ≤ p ≤ ∞, and derives a converse regularity statement.The argument uses interpolation and localized-kernel estimates.Source: §7 · p. 21
• The proof of Theorem 5.5 constructs nonnegative weights by extending a positive functional on the range of a sampling operator.The weights are then normalized over partition cells to obtain the stated construction.Source: §7 · p. 23
• Quadrature exactness alone does not imply σ2L(τ; P) = P when products from Π2L need not lie in Π2A∗L.The proof instead uses Lemma 7.4 to show σ2L(τ; P) is close to P.Source: §7 · p. 24

A Appendix

• The appendix states that proving this product property verifies the product assumption for eigenfunctions of the specified operators under technical conditions.It follows an idea from Geller and Pesenson.Source: A Appendix · pp. 24–25
• For the specified second-order, strongly uniformly elliptic self-adjoint operators, products QR of elements of ΠL belong to ΠA∗L for some A∗ ≥ 2.The conclusion assumes conditions (A.1) and (A.2).Source: A Appendix · p. 25
• The proof combines Leibniz’s formula for derivatives with Parseval’s identity to establish the product property.It concludes by letting the derivative-order parameter N tend to infinity.Source: A Appendix · pp. 25–26

Background

• Classical MZ inequalities compare integrals and weighted samples of trigonometric polynomials, with endpoint bounds requiring more sample points for the lower inequality.These inequalities are also called large sieve or network inequalities and have applications in approximation theory, number theory, and signal processing.Source: §1 · p. 29

Applications

• For kernel-based approximation, the MZ inequality at p = 1 implies that the continuous and discretized operator norms have the same order of magnitude as L grows.This applies to hyperinterpolation on the sphere, where the kernel is the reproducing kernel for spherical polynomials; the weights need not be nonnegative.Source: §1 · p. 30
• For randomly chosen nodes, the p = ∞ inequality can turn bounds on a kernel-weight sum into probability bounds for the discretized operator norm.The passage describes this as an application of probabilistic estimates for fixed x.Source: §1 · p. 30
• For least-squares approximation, the p = 2 inequality bounds the Gram matrix eigenvalues by the norm-comparison constants, with a ratio nearer 1 giving better conditioning.The Gram matrix entries are weighted sums of products of basis functions.Source: §1 · p. 30
• When the p = 1 constants are sufficiently close to 1, the inequalities lead to positive quadrature formulas exact on Π_cL for some constant c.The authors report this result across a number of their papers.Source: §1 · p. 30

Main contributions

• The paper gives conditions for MZ inequalities across 1 ≤ p ≤ ∞, including constants arbitrarily close to 1, and intrinsic characterizations of sampling systems for inequalities without that sharp-constant requirement.The framework includes weighted averages over balls, and uses a new interpolation result for the sharp form at intermediate p.Source: §1 · p. 31

Manifolds

• A differentiable manifold is described by compatible coordinate charts covering the space, and may be viewed intuitively as a surface without requiring an ambient Euclidean space.The paper assumes the manifold is Hausdorff and has a countable basis.Source: §2 · p. 32

Riemannian structure

• A smoothly varying inner product on tangent spaces defines a Riemannian metric, which determines curve lengths and the volume element; shortest joining curves are geodesics.The geodesic distance is based on these geodesics.Source: §2 · p. 32

Differential operators

• The Laplace–Beltrami operator is elliptic, so on a compact connected manifold it has a discrete spectrum and an orthonormal system of eigenfunctions.The section also reviews gradients and divergence as differential operators on the manifold.Source: §2 · p. 33

Setting

• The setting uses diffusion polynomials ΠL formed from orthonormal functions whose nondecreasing frequency parameters ℓk tend to infinity.Geodesic balls are defined using ρ, the geodesic distance.Source: §3 · p. 33

Heat-kernel assumptions

• The heat kernel is assumed to have Gaussian-type decay in distance, with a corresponding decay bound for its gradient.The stated bounds apply for t ∈ (0, 1], and the gradient is taken with respect to y.Source: §3 · p. 33

Bessel-system assumption

• The assumptions include that each family {Fφk}, for a vector field F, is a Bessel system; orthonormal systems themselves are examples of Bessel systems.The paper gives derivatives of Laplace–Beltrami eigenfunctions as an example and specifies a corresponding Bessel bound.Source: §3 · p. 34

Consequences

• The stated assumptions imply a homogeneity bound for μ on geodesic balls, comparing μ(B(x,R)) with μ(B(x,r)) when 0 < r ≤ 1 and R > 0.The bound scales by a constant times (R/r)^α.Source: §3 · p. 34

Measure formulation

• Writing discretized sums as integrals against a measure accommodates point samples as well as weighted averages over small balls.This notation also handles functions defined only almost everywhere and weighted-manifold measures.Source: §4 · p. 35

Signed measures

• For signed measures, total variation captures the magnitude of the weights, and the support consists of points whose every ball has positive total variation measure.For point masses with weights wy, the total variation is the sum of |wy|.Source: §4 · p. 35

MZ measures

• An MZ measure is one for which the constants in the norm-comparison inequalities do not depend on L.The inequalities are expressed using the measure formulation of the discretization.Source: §4 · pp. 35–36

Uniformity in L

• For each finite-dimensional ΠL, norm-comparison inequalities hold with constants that may depend on L; the section focuses on conditions making them uniform in L.The section also introduces Nikolskii inequalities as a property of diffusion polynomials.Source: §4 · pp. 35–36

Theorem 5.1

• The section also states a sharp MZ inequality theorem for finite node sets and discusses positive quadrature formulas.Part (a) has an earlier version for p=1 [11, Theorem 3.2]; the supplied theorem statement does not include its full hypotheses or bounds.Source: §5 · pp. 37, 39

Theorem 5.2

• Theorem 5.2 constructs a partition with cells near selected support points, cell measure comparable to d^α, and positive lower bounds on |ν|(Yk).The construction supplies the partition used to extend the MZ inequalities to general measures.Source: §5 · p. 37

Theorem 5.3

• Theorem 5.3 gives MZ inequalities for signed measures under scale conditions on d, L, and η; its p=∞ case settles that endpoint.For 1≤p<∞, the theorem's conditions include L≤c2η(pd)^-1; the paper then focuses on this finite-p range.Source: §5 · p. 38

Theorem 5.4

• For 1≤p<∞, norm domination by ν implies 1/L-regularity, while regularity and dominance conditions give the corresponding norm inequalities and converse criteria.The converse dominance criterion requires ν to be 1/L-regular and S>α an integer.Source: §5 · p. 39

Theorem 5.5

• If δ(supp(ν))<d and L is bounded by a constant times d^-1, a nonnegative simple function W on supp(ν) exists satisfying the stated quadrature condition.The authors note that the conditions for (5.17) depend only on supp(ν).Source: §5 · p. 40

Theorem 5.6

• Under the product assumption, positive quadrature measures of order 2A∗L yield the stated norm comparison for ΠL, with constants independent of τ and L.The assumption is conjectured to hold for every analytic manifold.Source: §5 · p. 40

6 Preparatory results

• The section assembles interpolation, localized polynomial operator, and Krein extension tools for the proofs in Section 5.It also summarizes properties of localized kernels and diffusion polynomials.Source: §6 · p. 40

6.1 Riesz–Thorin interpolation theorem

• The Riesz–Thorin theorem interpolates operator bounds between endpoint spaces when 1/p and 1/r vary linearly between endpoint exponents.The authors include a proof in the form they need, following the usual theorem [33, Chapter XII, Theorem 1.11].Source: §6.1 · p. 41

6.2 Localized polynomial operators

• Localized polynomial operator estimates provide Lp analogues of earlier bounds, including bounds for P and its gradient on node-centered neighborhoods.The stated estimates depend on δ(C)L and the localization exponent S.Source: §6.2 · pp. 43–44

6.3 Krein’s extension theorem

• The Krein extension result extends a functional positive on V with respect to K to a positive functional on X while preserving a norm bound.The bound is stated in Theorem 6.3 under its additional assumptions.Source: §6.3 · p. 45

Proof of Theorem 5.2

• Theorem 5.2’s partition is built through successive refinements that control cell measure, |ν|-mass, and inclusion of selected support points.The resulting cells satisfy µ(Yk)∼d^α and contain their selected nodes.Source: §7 · pp. 47–48

Proof of Theorem 5.4

• The proof of Theorem 5.4 uses interpolation to extend endpoint norm bounds to all p, then derives regularity from norm domination.The converse uses localized kernels to obtain a lower bound on measure in balls of radius proportional to 1/L.Source: §7 · p. 49

Proof of Theorem 5.5

• For Theorem 5.5, the proof applies positive-functional extension to obtain nonnegative cell weights and defines W by dividing each weight by |ν|(Yk).For d-regular ν, the resulting W has a lower bound proportional to |||ν|||R,d^-1.Source: §7 · p. 51

Proof of Theorem 5.6

• The proof of Theorem 5.6 uses the product assumption and quadrature exactness to bound local mass, then applies Theorem 5.4 to obtain the norm comparison.The local mass bound is τ(B(x,β/L))≤c4L^-α.Source: §7 · p. 51

Appendix

• For eigenfunctions of a second-order, strongly uniformly elliptic self-adjoint operator satisfying (A.1) and (A.2), QR∈ΠA∗L whenever Q,R∈ΠL.The appendix presents this as a stronger statement establishing the product assumption.Source: A Appendix · p. 53