Int J Geomath (2012) 3:95–117 DOI 10.1007/s13137-011-0029-7 ORIGINAL PAPER Filtered hyperinterpolation: a constructive polynomial approximation on the sphere Ian H. Sloan · Robert S. Womersley Received: 11 December 2011 / Accepted: 30 December 2011 / Published online: 20 January 2012 © Springer-Verlag 2012 Abstract This paper considers a fully discrete filtered polynomial approximation on the unit sphere S d . For f ∈ C (S d ), V (a) L , N f is a polynomial approximation which is exact for all spherical polynomials of degree at most L , so it inherits good con- vergence properties in the uniform norm for sufficiently smooth functions. The oscil- lations 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 =⌈aL ⌉+ L - 1. The main theoretical result is that the uniform norm ‖V (a) L , N ‖ of the filtered hyperinterpolation operator is bounded independently of L , providing both good convergence and stabil- ity properties. Numerical experiments on 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 I. H. Sloan · R. S.Womersley (B ) School of Mathematics and Statistics, University of New South Wales, Sydney 2052, Australia e-mail: R.Womersley@unsw.edu.au I. H. Sloan e-mail: I.Sloan@unsw.edu.au 123