Riemann localisation on the sphere

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.