Riemann Localisation on the Sphere

Research output: Journal Publications and Reviews (RGC: 21, 22, 62)21_Publication in refereed journalpeer-review

4 Scopus Citations
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Author(s)

  • Yu Guang Wang
  • Ian H. Sloan
  • Robert S. Womersley

Related Research Unit(s)

Detail(s)

Original languageEnglish
Pages (from-to)141-183
Journal / PublicationJournal of Fourier Analysis and Applications
Volume24
Issue number1
Online published1 Aug 2016
Publication statusPublished - Feb 2018

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 SdRd+1, d ≥ 2, we mean that for a suitable subset X of L(Sd), 1 ≤ p ≤ ∞, the Lp-norm of the Fourier local convolution of ƒX converges to zero as the degree goes to infinity. The Fourier local convolution of ƒ at xSd 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.

Research Area(s)

  • Filtered polynomial approximation, Riemann-Lebesgue lemma, Localization, Dirichlet kernel, Jacobi weights, POINTWISE FOURIER INVERSION, EIGENFUNCTION-EXPANSIONS, COMPACT MANIFOLDS, PINSKY PHENOMENON, WAVELETS, NEEDLETS, FRAMES

Citation Format(s)

Riemann Localisation on the Sphere. / Wang, Yu Guang; Sloan, Ian H.; Womersley, Robert S.

In: Journal of Fourier Analysis and Applications, Vol. 24, No. 1, 02.2018, p. 141-183.

Research output: Journal Publications and Reviews (RGC: 21, 22, 62)21_Publication in refereed journalpeer-review