PreprintCC BY 4.0arXiv
The MZ-property for three-term differences
Abstract
We study the Marcinkiewicz--Zygmund property for finite differences. A difference D(f,h)=∑k=1n ak f(bkh) is said to have the MZ-property if, for every measurable function f, D(f,h)=o(h) and f(h)=o(1) imply f(h)=o(h). In this paper, we characterize all three-node differences D(f,h)=af(qh)+bf(rh)+cf(sh) having the MZ-property. In particular, for irrational differences with distinct nodes, we obtain explicit necessary and sufficient conditions in terms of the coefficients a,b,c and the nodes q,r,s. As an application, our results resolve an open problem concerning a second-order Riemann difference.
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BibTeX
@misc{fejzic2026mzproperty,
title = {{The MZ-property for three-term differences}},
author = {Hajrudin Fejzić and Chris Freiling},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01406},
eprint = {2611.01406},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01406},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The MZ-property for three-term differences
AU - Hajrudin Fejzić
AU - Chris Freiling
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01406
UR - https://arxiv.org/abs/2611.01406
AB - We study the Marcinkiewicz--Zygmund property for finite differences. A difference D(f,h)=∑_k=1^n a_k f(b_kh) is said to have the MZ-property if, for every measurable function f, D(f,h)=o(h) and f(h)=o(1) imply f(h)=o(h). In this paper, we characterize all three-node differences D(f,h)=af(qh)+bf(rh)+cf(sh) having the MZ-property. In particular, for irrational differences with distinct nodes, we obtain explicit necessary and sufficient conditions in terms of the coefficients a,b,c and the nodes q,r,s. As an application, our results resolve an open problem concerning a second-order Riemann difference.
ER -