PreprintarXiv licencearXiv
Rudin's Problem on Uniformly Bounded Homogeneous Orthonormal Bases
Abstract
Rudin's problem asks whether, in the unit ball of ℂd with d≥2 fixed, the Hardy space H2(∂𝔹d) admits a uniformly bounded orthonormal basis consisting of homogeneous holomorphic polynomials. For d=2 and d=3 the answer is due to Bourgain, but the general-dimensional case has remained open. This paper gives an affirmative answer to Rudin's problem in every fixed dimension. More precisely, we prove that there is an orthonormal basis consisting of homogeneous polynomials {pj}j=1∞ of H2(∂𝔹d) such that |pj|L∞(∂𝔹d)≤ exp(1/2dlog d+O(d)). Here the implicit constant in O(d) is independent of d and j.
Related
- Uniformly Bounded Orthonormal Bases of Homogeneous Polynomials on the Complex Sphere
- Low-dimensional approximation of uniformly bounded orthonormal systems
- A Hierarchical Online-Compression Approach to Proportional Subsets of Bounded Orthonormal Systems
- The solution to Kadison's problem on orthonormal bases of unitaries for type II1 factors
- Relative property (T) for uniformly bounded representations
BibTeX
@misc{wang2026rudins,
title = {{Rudin's Problem on Uniformly Bounded Homogeneous Orthonormal Bases}},
author = {Penghui Wang and Fan Zhang and Zeyou Zhu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29946},
eprint = {2610.29946},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29946},
}
RIS
TY - UNPB
TI - Rudin's Problem on Uniformly Bounded Homogeneous Orthonormal Bases
AU - Penghui Wang
AU - Fan Zhang
AU - Zeyou Zhu
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29946
UR - https://arxiv.org/abs/2610.29946
AB - Rudin's problem asks whether, in the unit ball of ℂ^d with d≥2 fixed, the Hardy space H^2(∂𝔹_d) admits a uniformly bounded orthonormal basis consisting of homogeneous holomorphic polynomials. For d=2 and d=3 the answer is due to Bourgain, but the general-dimensional case has remained open. This paper gives an affirmative answer to Rudin's problem in every fixed dimension. More precisely, we prove that there is an orthonormal basis consisting of homogeneous polynomials {p_j}_j=1^∞ of H^2(∂𝔹_d) such that |p_j|_L^∞(∂𝔹_d)≤ exp(1/2dlog d+O(d)). Here the implicit constant in O(d) is independent of d and j.
ER -