Riemann localisation on the sphere

TL;DR

This paper investigates the Riemann localisation property for Fourier-Laplace series on spheres, showing it holds for 2D but not higher dimensions unless a filtered approach is used, with results based on kernel estimates.

Contribution

It demonstrates the dimension-dependent validity of Riemann localisation on spheres and introduces filtered kernels to restore localisation in higher dimensions.

Findings

Riemann localisation holds for 2D spheres with smooth functions.

Localisation fails for spheres of dimension greater than 2 without filtering.

Filtered kernels restore localisation property in higher dimensions.

Abstract

This paper first shows that the Riemann localisation property holds for the Fourier-Laplace series partial sum for sufficiently smooth functions on the two-dimensional sphere, but does not hold for spheres of higher dimension. By Riemann localisation on the sphere $\mathbb{S}^{d}\subset\mathbb{R}^{d+1}$, $d\ge2$, we mean that for a suitable subset $X$ of $\mathbb{L}_{p}(\mathbb{S}^{d})$, $1\le p\le \infty$, the $\mathbb{L}_{p}$-norm of the Fourier local convolution of $f\in X$ converges to zero as the degree goes to infinity. The Fourier local convolution of $f$ at $\boldsymbol{x}\in\mathbb{S}^{d}$ is the Fourier convolution with a modified version of $f$ obtained by replacing values of $f$ by zero on a neighbourhood of $\boldsymbol{x}$. The failure of Riemann localisation for $d>2$ can be overcome by considering a filtered version: we prove that for a sphere of any dimension and sufficiently smooth filter the corresponding local convolution always has the Riemann localisation property. Key tools are asymptotic estimates of the Fourier and filtered kernels.