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Optimal polynomial meshes beyond geometric boundary regularity via Green sublevels
Abstract
We establish a potential-theoretic sufficient condition for a compact subset of the real line to admit an optimal polynomial mesh. The criterion bounds the cardinality of a norming set by a linear term in the polynomial degree plus the number of connected components of a Green sublevel at height α/n, where α>0 is fixed. By combining this principle with an estimate due to Andrievskii, we prove that every uniformly perfect compact subset of ℝ admits an optimal polynomial mesh. A standard product argument then produces optimal meshes on finite Cartesian products; in particular, it yields an optimal mesh on the planar Cantor dust C× C, which is self-similar, totally disconnected, and has empty interior.
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BibTeX
@misc{pinasco2026optimal,
title = {{Optimal polynomial meshes beyond geometric boundary regularity via Green sublevels}},
author = {Damián Pinasco and María Victoria Venuti},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.27618},
eprint = {2608.27618},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.27618},
}
RIS
TY - UNPB
TI - Optimal polynomial meshes beyond geometric boundary regularity via Green sublevels
AU - Damián Pinasco
AU - María Victoria Venuti
PY - 2026
DA - 2026/08/27
PB - arXiv
DO - 10.48550/arxiv.2608.27618
UR - https://arxiv.org/abs/2608.27618
AB - We establish a potential-theoretic sufficient condition for a compact subset of the real line to admit an optimal polynomial mesh. The criterion bounds the cardinality of a norming set by a linear term in the polynomial degree plus the number of connected components of a Green sublevel at height α/n, where α>0 is fixed. By combining this principle with an estimate due to Andrievskii, we prove that every uniformly perfect compact subset of ℝ admits an optimal polynomial mesh. A standard product argument then produces optimal meshes on finite Cartesian products; in particular, it yields an optimal mesh on the planar Cantor dust C× C, which is self-similar, totally disconnected, and has empty interior.
ER -