The Marcinkiewicz-Zygmund Property for Riemann Differences with Geometric Nodes
Hajrudin Fejzić
TL;DR
This work provides a complete analytic criterion for when geometric-node Riemann differences possess the Marcinkiewicz-Zygmund property by reducing the problem to a recurrence $D(h)=R(qh)-A R(h)$ and a sharp annulus-based root-location condition. The authors prove a precise trichotomy: if $|A|$ lies outside the critical annulus determined by $|q|$ and $n$, then $R(h)=o(h^n)$ follows from $D(h)=o(h^n)$ and $R(h)=o(h^{n-1})$, while inside the annulus counterexamples exist. This leads to a complete classification of all geometric-node Riemann differences with the MZ property and extends to broader generalized differences via the characteristic polynomial framework and even/odd symbol decompositions for real-valued cases. The results not only settle questions about classical forward and symmetric templates but also reveal that MZ behavior is governed by root geometry rather than node symmetry, with practical implications for constructing MZ-compatible differences and understanding when continuity can restore differentiability in borderline cases.
Abstract
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.