Sharp Norm Approximation of Covering Markov Operators
Abstract
We study the operator-norm approximation of Markov operators associated with finite graph coverings. Let q:Y→ X be a connected covering of degree d≥ 2 between finite connected graphs, and let Aq:C(Y)→ C(X) be its fiber-averaging operator. We show that Aq is not a finite convex combination of extreme Markov operators, although such combinations approximate Aq arbitrarily well in operator norm. We determine the exact distances from Aq to convex combinations of at most m extreme Markov operators, with arbitrary or equal convex weights. If m=kd+r, where k≥ 0 and 1≤ r≤ d, these distances are 2(d−r+1)/[d(k(d−r+1)+1)] and 2r(d−r+1)/(dm), respectively. Both distances are attained and depend only on the covering degree d and the number of terms m. We also determine when equal weights are optimal and quantify the additional approximation error when they are not. Our proofs use total-variation estimates and continuous rearrangements of labelled weighted atoms. In particular, the unrestricted errors have a common scaled limit, while the equal-weight errors exhibit residue-dependent asymptotic behavior.
Related
- Covering lemmas associated to maximal operators of weak type
- Approximation in norm and energy with bounded functions for autonomous variational problems
- A sharp covering theorem and Solyanik estimates for Euclidean balls
- A Sharp Product Bound for Disjoint Cross-Intersecting 3-Graphs with Covering Number Three
- Weighted composition--differentiation operators on mixed-norm, derivative Hardy and F(p,q,s) spaces
BibTeX
@misc{duru2026sharp,
title = {{Sharp Norm Approximation of Covering Markov Operators}},
author = {Hülya Duru and Serkan İlter and Aygül Bilgin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35575},
eprint = {2610.35575},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35575},
}
RIS
TY - UNPB
TI - Sharp Norm Approximation of Covering Markov Operators
AU - Hülya Duru
AU - Serkan İlter
AU - Aygül Bilgin
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.35575
UR - https://arxiv.org/abs/2610.35575
AB - We study the operator-norm approximation of Markov operators associated with finite graph coverings. Let q:Y→ X be a connected covering of degree d≥ 2 between finite connected graphs, and let A_q:C(Y)→ C(X) be its fiber-averaging operator. We show that A_q is not a finite convex combination of extreme Markov operators, although such combinations approximate A_q arbitrarily well in operator norm. We determine the exact distances from A_q to convex combinations of at most m extreme Markov operators, with arbitrary or equal convex weights. If m=kd+r, where k≥ 0 and 1≤ r≤ d, these distances are 2(d−r+1)/[d(k(d−r+1)+1)] and 2r(d−r+1)/(dm), respectively. Both distances are attained and depend only on the covering degree d and the number of terms m. We also determine when equal weights are optimal and quantify the additional approximation error when they are not. Our proofs use total-variation estimates and continuous rearrangements of labelled weighted atoms. In particular, the unrestricted errors have a common scaled limit, while the equal-weight errors exhibit residue-dependent asymptotic behavior.
ER -