Explicit Measurable Counterexamples to Finite-Difference Operators
Abstract
Let D(f,h)=∑k=1n ak f(bkh) 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) Longrightarrow 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 for every h∈ℂ, while, in fact, f(h)not=o(1) as h→0.
Related
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BibTeX
@misc{fejzic2026explicit,
title = {{Explicit Measurable Counterexamples to Finite-Difference Operators}},
author = {Hajrudin Fejzić},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.30583},
eprint = {2610.30583},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.30583},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Explicit Measurable Counterexamples to Finite-Difference Operators
AU - Hajrudin Fejzić
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.30583
UR - https://arxiv.org/abs/2610.30583
AB - Let D(f,h)=∑_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) Longrightarrow 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 for every h∈ℂ, while, in fact, f(h)not=o(1) as h→0.
ER -