Uniformly bounded orthonormal polynomials on the sphere

TL;DR

This paper constructs large orthonormal systems of uniformly bounded polynomials on spheres and sections of line bundles on Kähler manifolds, nearly matching the full dimension of the respective polynomial spaces.

Contribution

It introduces a method to build nearly complete orthonormal systems of bounded polynomials and sections on complex manifolds, extending previous results.

Findings

Constructed orthonormal systems with size > (1 - ε) times the dimension

Systems are uniformly bounded in supremum norm

Applicable to spheres and positive holomorphic line bundles

Abstract

Given any $\varepsilon>0$, we construct an orthonormal system of $n_k$ uniformly bounded polynomials of degree at most $k$ on the unit sphere in $\mathbb R^{m+1}$ where $n_k$ is bigger than $1-\varepsilon$ times the dimension of the space of polynomials of degree at most $k$. Similarly we construct an orthonormal system of sections of powers $L^k$ of a positive holomorphic line bundle on a compact K\"ahler manifold with cardinality bigger than $1-\varepsilon$ times the dimension of the space of global holomorphic sections to $L^k$.