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Filtered hyperinterpolation: a constructive polynomial approximation on the sphere

  • Ian H. Sloan
  • Robert S. WomersleyEmail author
Original Paper

Abstract

This paper considers a fully discrete filtered polynomial approximation on the unit sphere \({\mathbb{S}^{d}.}\) For \({f \in C(\mathbb{S}^{d}),V_{L,N}^{(a)} \, f}\) is a polynomial approximation which is exact for all spherical polynomials of degree at most L, so it inherits good convergence properties in the uniform norm for sufficiently smooth functions. The oscillations often associated with polynomial approximation of less smooth functions are localised by using a filter with support [0, a] for some a > 1, and with the value 1 on [0, 1]. The allowed choice of filters includes a recently introduced filter with minimal smoothness, and other smoother filters. The approximation uses a cubature rule with N points which is exact for all polynomials of degree \({t = \left\lceil{a L}\right\rceil+L-1.}\) The main theoretical result is that the uniform norm \({\|V_{L,N}^{(a)} \|}\) of the filtered hyperinterpolation operator is bounded independently of L, providing both good convergence and stability properties. Numerical experiments on \({\mathbb{S}^{2}}\) with a variety of filters, support intervals and cubature rules illustrate the uniform boundedness of the operator norm and the convergence of the filtered hyperinterpolation approximation for both an arbitrarily smooth function and a function with derivative discontinuities.

Keywords

Polynomial approximation Filter Sphere Lebesgue constant Hyperinterpolation 

Mathematics Subject Classification (2000)

65D10 86A99 41A30 41A63 

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Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsUniversity of New South WalesSydneyAustralia

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