Nearly tight framelet systems from nested Marcinkiewicz--Zygmund measures on compact Riemannian manifolds

Abstract

Tight framelet systems on manifolds provide exact energy preservation and one-pass reconstruction, but their standard semi-discretization relies on polynomial-exact quadrature rules, which are difficult to reconcile with scattered and progressively refined data. We develop nested Marcinkiewicz--Zygmund (MZ) measures on compact Riemannian manifolds as a quantitative alternative. Using dyadic quasi-uniform nested point sets and local partition weights, we establish a sequence of deterministic MZ inequalities for diffusion polynomial spaces. The result has two complementary forms: a fixed-tolerance version, in which the polynomial bandwidth grows dyadically with the level, and a fixed-bandwidth refinement version, in which the MZ tolerance improves as more nested nodes are added. When these measures are used to discretize continuous tight framelets, the MZ tolerance transfers directly to the deviation of the frame bounds from one. Thus quadrature exactness is relaxed with a controlled loss of tightness, and near-tightness improves along the same nested hierarchy. For the fully discrete setting, we develop filter-bank analysis and synthesis procedures, characterize one-pass and canonical reconstruction, and show that the MZ tolerance also controls the conditioning of the frame-operator equation. Fast implementation of filter-bank transforms is also presented. Numerical experiments on the sphere and the flat torus illustrate the resulting multiscale decompositions and reconstruction behavior.