PreprintCC BY 4.0arXiv
Uniformly Bounded Orthonormal Bases of Homogeneous Polynomials on the Complex Sphere
Abstract
We resolve the long-standing problem of Rudin concerning uniformly bounded orthonormal bases of homogeneous polynomials on complex unit spheres. Specifically, we show that for every K ≥ 2 and N ≥ 0 there exists a basis of degree N homogeneous polynomials in ℂK that is orthonormal with respect to the surface measure of the unit sphere of ℂK and uniformly bounded by a constant depending only on K on the unit ball of ℂK. Bourgain resolved the case K = 2 in 1985 and the case K = 3 in 2016. No cases K ≥ 4 were previously known.
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BibTeX
@misc{goncalves2026uniformly,
title = {{Uniformly Bounded Orthonormal Bases of Homogeneous Polynomials on the Complex Sphere}},
author = {Felipe Gonçalves and João P. Ferreira and Maksym Radziwiłł},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14160},
eprint = {2610.14160},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14160},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Uniformly Bounded Orthonormal Bases of Homogeneous Polynomials on the Complex Sphere
AU - Felipe Gonçalves
AU - João P. Ferreira
AU - Maksym Radziwiłł
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.14160
UR - https://arxiv.org/abs/2610.14160
AB - We resolve the long-standing problem of Rudin concerning uniformly bounded orthonormal bases of homogeneous polynomials on complex unit spheres. Specifically, we show that for every K ≥ 2 and N ≥ 0 there exists a basis of degree N homogeneous polynomials in ℂ^K that is orthonormal with respect to the surface measure of the unit sphere of ℂ^K and uniformly bounded by a constant depending only on K on the unit ball of ℂ^K. Bourgain resolved the case K = 2 in 1985 and the case K = 3 in 2016. No cases K ≥ 4 were previously known.
ER -