Zonal spherical polynomials with minimalL ₁-norm

Abstract

Radial functions have become a useful tool in numerical mathematics. On the sphere they have to be identified with the zonal functions. We investigate zonal polynomials with mass concentration at the pole, in the sense of their L₁-norm is attaining the minimum value. Such polynomials satisfy a complicated system of nonlinear equations (algebraic if the space dimension is odd, only) and also a singular differential equation of third order. The exact order of decay of the minimum value with respect to the polynomial degree is determined. By our results we can prove that some nodal systems on the sphere, which are defined by a minimum property, are providing fundamental matrices which are diagonal-dominant or bounded with respect to the ∞-norm, at least, as the polynomial degree tends to infinity.