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.
Categories: math.CA