标题： 黎曼差分的马尔钦凯维茨-齐格蒙德性质与几何节点 显示英文标题

标题： The Marcinkiewicz-Zygmund Property for Riemann Differences with Geometric Nodes

摘要： 我们研究当一个阶数为$ n $的黎曼差分具有 Marcinkiewicz-Zygmund (MZ) 性质的情况：即，条件$ f(h) = o(h^{n-1}) $和$ Df(h) = o(h^n) $是否能推出$ f(h) = o(h^n) $。已知对于一些具有几何节点的经典例子，如$ \{0, 1, q, \dots, q^{n-1}\} $和$ \{1, q, \dots, q^n\} $，该蕴含成立，从而提出了一个猜想，认为这些是唯一具有 MZ 性质的黎曼差分。然而，这一猜想被具有节点$ \{-1, 0, 1, 2\} $的三阶例子所驳斥，我们在此提供了进一步的反例和一个一般分类。 我们通过开发一个递归框架，建立了MZ性质的完整解析准则：我们分析当一个函数$ R(h) $满足$ D(h) = R(qh) - A R(h) $，以及$ D(h) = o(h^n) $和$ R(h) = o(h^{n-1}) $时，何时会强制$ R(h) = o(h^n) $。 我们证明，当且仅当$ A $位于由$ q $和$ n $确定的临界模环之外时成立，这涵盖了$ |q| > 1 $和$ |q| < 1 $的情况。 这导致了所有具有几何节点并具有 MZ 性质的黎曼差分的完整特征描述，并提供了一个适用于更广泛类别的广义差分的灵活分析框架。 显示英文摘要

摘要： We study when a Riemann difference of order $ n $ possesses the Marcinkiewicz-Zygmund (MZ) property: that is, whether the conditions $ f(h) = o(h^{n-1}) $ and $ Df(h) = o(h^n) $ imply $ f(h) = o(h^n) $. This implication is known to hold for some classical examples with geometric nodes, such as $ \{0, 1, q, \dots, q^{n-1}\} $ and $ \{1, q, \dots, q^n\} $, leading to a conjecture that these are the only such Riemann differences with the MZ property. However, this conjecture was disproved by the third-order example with nodes $ \{-1, 0, 1, 2\} $, and we provide further counterexamples and a general classification here. We establish a complete analytic criterion for the MZ property by developing a recurrence framework: we analyze when a function $ R(h) $ satisfying $ D(h) = R(qh) - A R(h) $, together with $ D(h) = o(h^n) $ and $ R(h) = o(h^{n-1}) $, forces $ R(h) = o(h^n) $. We prove that this holds if and only if $ A $ lies outside a critical modulus annulus determined by $ q $ and $ n $, covering both $ |q| > 1 $ and $ |q| < 1 $ cases. This leads to a complete characterization of all Riemann differences with geometric nodes that possess the MZ property, and provides a flexible analytic framework applicable to broader classes of generalized differences.