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 S^d ⊂ R^d+1, d ≥ 2, we mean that for a suitable subset X of L_p (S^d), 1 ≤ p ≤ ∞, the L_p-norm of the Fourier local convolution of ƒ ∈ X converges to zero as the degree goes to infinity. The Fourier local convolution of ƒ at x ∈ S^d is the Fourier convolution with a modified version of ƒ obtained by replacing values of ƒ by zero on a neighbourhood of 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.