Asymptotics of Weighted Bergman Polynomials under Finite Regularity
Abstract
We study strong asymptotics of planar orthogonal polynomials with respect to fixed weighted area measures on Jordan domains. The aim is to understand how the regularity of the domain and the weight enter the large-degree asymptotic behavior of these polynomials. For domains with real-analytic boundaries, Suetin established leading-order asymptotics for Hölder continuous weights, while Hedenmalm--Wennman obtained a full asymptotic expansion for real-analytic weights. We interpolate between these results while allowing both the boundary and the weight to have finite smoothness. The resulting expansions have a length determined by the available regularity and hold in the exterior region and, under sufficient regularity conditions, in a shrinking neighborhood of the boundary.
Related
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BibTeX
@misc{hubner2026asymptotics,
title = {{Asymptotics of Weighted Bergman Polynomials under Finite Regularity}},
author = {Lennart Aljoscha Hübner},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24693},
eprint = {2610.24693},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24693},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Asymptotics of Weighted Bergman Polynomials under Finite Regularity
AU - Lennart Aljoscha Hübner
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2610.24693
UR - https://arxiv.org/abs/2610.24693
AB - We study strong asymptotics of planar orthogonal polynomials with respect to fixed weighted area measures on Jordan domains. The aim is to understand how the regularity of the domain and the weight enter the large-degree asymptotic behavior of these polynomials. For domains with real-analytic boundaries, Suetin established leading-order asymptotics for Hölder continuous weights, while Hedenmalm--Wennman obtained a full asymptotic expansion for real-analytic weights. We interpolate between these results while allowing both the boundary and the weight to have finite smoothness. The resulting expansions have a length determined by the available regularity and hold in the exterior region and, under sufficient regularity conditions, in a shrinking neighborhood of the boundary.
ER -