Uniformly bounded orthonormal polynomials on the sphere

Abstract

ABSTRACT. Given any ε > 0, we construct an orthonormal system of n k uniformly bounded polynomials of degree at most k on the unit sphere in R m+1 where n k 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 L k of a positive holomorphic line bundle on a compact Kähler manifold with cardinality bigger than 1 − ε times the dimension of the space of global holomorphic sections to L k .