Year: 2026
Type: preprint
Abstract: Let \[ D(f,h)=\sum_{k=1}^n a_k f(b_kh) \] be a finite-difference operator, where the coefficients may be complex and the nodes are real. We study the asymptotic implication \[ D(f,h)=o(h)\quad\Longrightarrow\quad f(h)=o(h) \] for Lebesgue measurable functions. In 2017, Ash, Catoiu, and Csörnyei \cite{ACC} showed that this implication holds for every measurable function precisely for operators belonging to a very restricted class of finite differences. To establish that every operator outside the exceptional class admits a counterexample, Ash, Catoiu, and Csörnyei analyze an infinite system of linear equations. Their argument is therefore existential in nature and does not provide an explicit description of the counterexamples. By contrast, we give a substantially simpler and elementary construction based on zeros of finite Laurent polynomials and multiplicative characters on finitely generated multiplicative subgroups. This provides a direct recipe for constructing measurable functions \(f\) such that \[ D(f,h)=0\qquad\text{for every }h\in\mathbb C, \] while, in fact, \[ f(h)\not=o(1)\qquad\text{as }h\to0. \]
Source: arXiv (Cornell University)
Author: Fejzić, Hajrudin
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223363114