紧致黎曼流形上嵌套Marcinkiewicz-Zygmund测度的近紧框架系统
Nearly tight framelet systems from nested Marcinkiewicz--Zygmund measures on compact Riemannian manifolds
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中文总结 AI 辅助
该研究针对流形紧框架半离散化难以适配散乱细化数据的问题，提出紧致黎曼流形上的嵌套MZ测度，建立对应不等式并开发滤波器组流程，经数值实验验证其多尺度分解与重建效果。
中文摘要 AI 辅助
流形上的紧框架系统可实现精确能量保存与单步重建，但其标准半离散化依赖多项式精确求积规则，难以适配散乱且逐步细化的数据。我们在紧致黎曼流形上开发嵌套Marcinkiewicz-Zygmund（MZ）测度作为定量替代方案。利用二进拟均匀嵌套点集与局部划分权重，我们为扩散多项式空间建立了一系列确定性MZ不等式，该结果包含两种互补形式：固定容差版本中，多项式带宽随层级二进式增长；固定带宽细化版本中，MZ容差随嵌套节点增加而提升。将这些测度用于离散化连续紧框架时，MZ容差直接传递至框架界与1的偏差，因此求积精确性得到放松，同时紧度损失可控，且近紧度沿同一嵌套层级提升。针对全离散设置，我们开发了滤波器组分析与合成流程，表征了单步与标准重建，并证明MZ容差还可控制框架算子方程的条件数，同时给出了滤波器组变换的快速实现方案。在球面与平坦环面上的数值实验，验证了所得多尺度分解与重建行为。
英文摘要
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.
