Rudin’s uniform boundedness problem in higher dimensions

Ascertain whether there exist uniformly bounded orthonormal bases for the spaces of holomorphic homogeneous polynomials in the unit ball B_m ⊂ ℂ^m for m ≥ 4; equivalently, determine if the L^∞ norms of normalized spherical harmonics of degree D on S^{2m−1} remain uniformly bounded as D → ∞ for m ≥ 4.

Background

Rudin asked whether spherical harmonics (or holomorphic homogeneous polynomials) admit uniformly bounded orthonormal bases. Bourgain proved positive answers for m = 2,3 (S3,S5) via Rudin–Shapiro constructions, but higher m remains unresolved.

This problem lies at the intersection of complex analysis, harmonic analysis, and special function theory.

References

The same question in higher dimensions remains open. Specifically, it is not known if there exist uniformly bounded orthonormal bases for the spaces of holomorphic homogeneous polynomials in $\mathbb{B}_m$, the unit ball in $\mathbb{C}m$, for $m \geq 4$.

— Mathematical exploration and discovery at scale (2511.02864 - Georgiev et al., 3 Nov 2025) in Subsection “Rudin problem for polynomials” (Section 4.33)