  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.
