Uniformly bounded orthonormal polynomials on the sphere

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 Kähler manifold with cardinality bigger than $1-\varepsilon$ times the dimension of the space of global holomorphic sections to $L^k$.

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