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Journal of Fourier Analysis and Applications

, Volume 24, Issue 1, pp 141–183 | Cite as

Riemann Localisation on the Sphere

  • Yu Guang Wang
  • Ian H. Sloan
  • Robert S. Womersley
Article
  • 199 Downloads

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 \(\mathbb {S}^{d}\subset \mathbb {R}^{d+1}\), \(d\ge 2\), 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 \(\mathbf {x}\in \mathbb {S}^{d}\) 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.

Keywords

Filtered polynomial approximation Riemann–Lebesgue lemma Localization Dirichlet kernel Jacobi weights 

Mathematics Subject Classification

42C15 42A63 41A10 33C55 33C45 

Notes

Acknowledgments

The authors would like to thank Christian Gerhards and Leonardo Colzani for their discussion and comments on the convergence of the Fourier local convolution and the localisation principle. The authors also thank the anonymous referees for their comments on simplifying the proof of Theorem 3.2. This research was supported under the Australian Research Council’s Discovery Project DP120101816. The first author was supported under the University International Postgraduate Award (UIPA) of UNSW Australia.

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Yu Guang Wang
    • 1
    • 2
  • Ian H. Sloan
    • 1
  • Robert S. Womersley
    • 1
  1. 1.School of Mathematics and StatisticsUniversity of New South WalesKensingtonAustralia
  2. 2.Department of MathematicsCity University of Hong KongKowloon TongHong Kong

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