Riemann localisation on the sphere

2015/10/23 by Wang, Yu Guang, Sloan, Ian H., Womersley, Robert S.
#33C45 #33C55 #41A10 #42A63 #42C15 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1510.06834

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

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