Uniformly bounded orthonormal polynomials on the sphere

marzo, jordi; ortega-cerd`, joaquim;

Given any ε > 0, we construct an orthonormal system of nk uniformly bounded polynomials of degree at most k on the unit sphere in Rm+1 where nk is bigger than 1 − ε times the dimension of the space of polynomials of degree at most k. Similarly, we construct an orthonormal system of sections of powers Ak of a positive holomorphic line bundle on a compact K¨ahler manifold with cardinality bigger than 1 − ε times the dimension of the space of global holomorphic sections
to Ak.