Concentration estimates for band-limited spherical harmonics expansions via the large sieve principle

TL;DR

This paper develops bounds for how well band-limited spherical harmonics can concentrate on the sphere, using large sieve techniques and estimates of spherical harmonic coefficients.

Contribution

It introduces a large sieve inequality analogue for spherical harmonics and provides new bounds based on Nyquist density and zonal filter coefficients.

Findings

Derived upper bounds for concentration in spherical harmonics expansions.

Established an analogue of the classical large sieve inequality for spherical harmonics.

Provided estimates of spherical harmonics coefficients for zonal filters.

Abstract

We study a concentration problem on the unit sphere $\mathbb{S}^2$ for band-limited spherical harmonics expansions using large sieve methods. We derive upper bounds for concentration in terms of the maximum Nyquist density. Our proof uses estimates of the spherical harmonics coefficients of certain zonal filters. We also demonstrate an analogue of the classical large sieve inequality for spherical harmonics expansions.