Sign equidistribution of Legendre polynomials

TL;DR

This paper proves that as Legendre polynomial degrees increase, the positive and negative regions become equally distributed, and the method also addresses a symmetry conjecture for eigenfunctions on the sphere.

Contribution

It establishes sign equidistribution for Legendre polynomials and applies the proof technique to a symmetry conjecture for spherical eigenfunctions.

Findings

Ratio of positive to negative regions approaches one as degree increases

Proof method applicable to symmetry conjecture in spherical eigenfunctions

Supports the understanding of polynomial sign distribution in approximation theory

Abstract

We prove {\em sign equidistribution} of Legendre polynomials: the ratio between the lengths of the regions in the interval $[-1, 1]$ where the Legendre polynomial assumes positive versus negative values, converges to one as the degree grows. The proof method also has application to the symmetry conjecture for a basis of eigenfunctions in the sphere.