Riemann Localisation on the Sphere

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}), (dge 2), we mean that for a suitable subset X of (mathbb {L}_{p}(mathbb {S}^{d})), (1le ple infty ), the (mathbb {L}_{p})-norm of the Fourier local convolution of (fin X) converges to zero as the degree goes to infinity. The Fourier local convolution of f at (mathbf {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 (mathbf {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.