Uniformly bounded orthonormal polynomials on the sphere

Given any $\epsilon >0$, 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 $1-\epsilon$ 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 $1-\epsilon$ times the dimension of the space of global holomorphic sections to $L^k$.